It is true though that unlike the Dutch settlers in the Cape Colony, the major Russian interest in North America was the fur trade, and therefore like the French in New France, they concentrated on cultivating relations with the Native peoples rather than shoving them aside for settlements. So the pattern of Russo-Native relations would be more New France, less British colony or Cape Colony.
Now I have heard of uprisings against the Russians happening so I don't suppose it was all sweetness and light. And what the Russians needed to enable survival first, and persistence against British and Yankee desires, as well as conceivable Spanish strong reactions, would be to have some land heavily settled agricultural land. The fur trade managers were always short of food! I don't understand why, in seeking to develop a food plantation, they skipped right past both Puget Sound and the Colombia river mouth and went most of the way to San Francisco Bay but not quite! (The last bit is easiest to understand--the Spanish had arrived some time before, and their Mexican successors stood guard on the Bay itself. The question is not so much why Fort Ross and not the fertile inland of California, but why skip Washington and Oregon? I've never understood that part!
So figuring they don't miss the much closer and also less defended and quite adequately large agricultural prospects, the next thing they need to do is settle them since native peoples are not going to cultivate nearly enough. But just to support the fur trader management (the major work of fur trading was done by the Natives themselves of course) would really not take a lot of land.
That's good because every acre they take is a grievance with Native people and the main business model of the Company is not to have those. As said though not all was not joy and harmony; part of that was I believe a certain amount of divide and rule, with the Russians exploiting existing conflicts between peoples. A canny enough expedition captain would identify the land the settlers would need, determine which Native peoples live there, and who their enemies are, and befriend the latter! Also find out something else beside the desired land these enemies desire in reward and scheme to get that too.
The point being that because the Russian colonies had a pretty light footprint, they did not require large numbers of colonists, and even if they had the controlled agricultural land they wanted it wouldn't be all that many peasants they'd need to settle, nor would disruption of Native peoples be nearly as large a problem as with Anglo style aggressive settlement and expansion.
The big issue is timing. Just when would some Russian decide that they need an agricultural colony in America to support the fur traders? The earlier the better. However, I believe the major reason that the Spanish decided to support Junipero Serra's drive to extend missions up the California coast was precisely their concern that the Russians were present far to the north; it was a preemptive act. They had had bases on the northwest Pacific coast of Mexico for a long time before these moves were made, starting just before the American Revolution. The last missions in the Bay Area were actually established under Mexican auspices, not Spanish. But as I understand it the Russians, though present enough to alarm the Spanish viceroys, were very very slow to move into Alaska, and it occurs to me that by the time the Fort Ross venture was undertaken, events had moved fast in the northwest, at least on paper. With the Louisiana Purchase, the United States suddenly had claims north of Spanish claims, which I believe were at some point settled at the current latitude of California and Nevada northern state borders. But just when? Before Mexico's crisis with Fort Ross, or after? I suspect that border was decided in the process of persuading the Russians to leave, and that the reason the Russians leapfrogged thousands of miles down the coast past both Puget Sound and the Columbia mouth was that by the time they hoped to set up a food colony, the British and Americans were already present on the ground disputing that zone with each other.
Russia was a nominal ally of the USA from the early days of the Revolutionary War when John Adams was delegated to Saint Petersburg to negotiate it right up to the Bolshevik Revolution OTL. ITTL, unless the Yankees are completely expelled from the Pacific coast by British maneuvers, I expect that to change, but also to hold for a while. The alliance was rather tenuous and lasted as long as it did OTL mainly because we had few to no conflicts of interest with the Russians, whereas we had mutual foes or potential foes. A strong Russian presence in the Pacific Northwest though, combined with the USA managing to get some salient onto the Pacific (almost certainly the Columbia river mouth, "Oregon" for some value of Oregon) would change the situation somewhat. However both nations would continue to have conflicts with Britain so the alliance might last, provided diplomats from both sides negotiated a viable modus viviendi between the two groups of settlers. If the Russians continue the pattern of thin settlement, the minimum needed to feed the fur traders, and alliance with Native peoples on the New France model, it might get dicy since Americans are going to start feeling confined in just Oregon and looking enviously on lands they regard as underutilized by "feckless" Native people under the Tsar's protection. And other Natives who are attacked and displaced by aggressive Yankees would flee to Russian lands and stir up a sentiment of hostility among the Native subjects of the Tsar. In these circumstances a populist grass-roots observation that Russia is among the worst possible examples of royal absolutist tyranny the Declaration of Independence denounced so loftily will tend to start poisoning the OTL good relations and greed will provide the driving force behind the wedge of democratic ideology. Not to mention racism of the kind that says Native Americans are people in the way.
If the Russians can move early enough, with some grand strategist of the profitable if marginal fur trade observing that a food colony is needed long before British and American claims start getting staked, and a systematic investigation of possibilities as near to the Alaskan fur source zone is undertaken, I suppose that when they reach Puget Sound they'll stop right there. There is plenty of land good enough for Russian farmers to produce ample products for the numbers of traders envisioned. I don't foresee any political difficulties in getting a calculated number of Russian settlers to go there, though the logistics of moving them all the way across Siberia to Pacific ports and thence across the ill-explored northern Pacific to Vancouver island (surely in this scenario with some Russian name) and thence to choice spots either on the island, or on the mainland (or on the small islands of the Sound) to settle and farm would be daunting. It would take years, I'd guess we have to allow a 15 year window between someone making the decision and the settlement being solid and viable and producing as planned. Those years should not overlap deadlines like the OTL expeditions of Astor (or earlier British ventures if any) or the Louisiana purchase by more than a few. Indeed such a Russian presence would mean that the late 18th and early 19th century would be quite different in the region, Cook and Vancouver's expeditions there, if not butterflied away, would have a different story and character and outcome.
Would the Russians reach for maximum expansion? I'd say no they would not. Their purpose is limited, and so are their effective resources in the Pacific. If they can secure the mouth of the Puget Sound they won't want to go further. The Columbia River remains up for grabs--the Russians might regret this later but I don't seem them going for maximal preemption on the thin margin they have.
If they were to do that by the way, it makes much more sense for them to push southward along the coast toward Mexico for their buffer than eastward inland. Indeed the policy of creating an informal buffer by supporting and protecting Native people largely undisturbed except for some supporting forts is a good one and characteristic of the OTL Russian policy in America, but I agree that it would hardly be practical to push it farther east than the Continental divide. But I don't see them moving past the Puget Sound region fast enough to stop US claims on Oregon.
So here's the best case scenario:
1) long before the US gets its claim to the coast with the LP, and before the British can get any settlements started, the Russians move into the Sound region, securing Vancouver Island and claiming all down the Sound with indefinite, vague claims south and east. They also go up the rivers into the interior.
2) best case for US relations--when Jefferson claims to the coast with Lewis and Clark, the US government and Tsarist one, valuing their tentative alliance (not sure about that with Jefferson in office, he was pro-French, but I believe the Tsar was vaguely aligned with Napoleon at the time, though obviously not a decade later!) cordially draw firm lines on the map. I believe the Americans will insist on claims on both sides of the Columbia, but I could see Jefferson agreeing to designate the north bank territory an Indian reserve, which the USA can fortify but not settle. This helps the Russian policy of a similar pattern of settled core surrounded by Native ally buffer and insulates the American settlement from direct contact with the Tsarist claimed border, which I suppose would run on the watershed between the Columbia and the Puget Sound river systems, to be determined by survey--since it lies in an Indian-inhabited neutral zone, it is not so critical as long as Russia and USA remain at peace.
3) the Americans, though feeling constrained and resentful and probably eventually encroaching on the Indian reserve in the north, have their attention diverted south to California. I don't want to game that out!
4) the Russians, feeling they need some counterweight to the strength of US settlement and the changing politics of American attitudes to their regime, redouble settling the Sound region. They need to support larger garrisons of soldiers (who can be recruited from settler peasants to an extent, and Russified Natives--also I expect intermarriage between Natives and Russians which produces a Metis population, with strong ties to Native societies but also to Russian, and genetically resistant to Eurasian disease). But a certain number of troops from Russia itself need to be included to reinforce loyalties.
By the way I expect the Russian peasant settlers to be pretty happy; they will be working better land than they are probably used to, and have wider prospects than most Russians of their class. They will definitely be loyal to Tsarist calls for defensive military service, and reliable. Their children and grandchildren will be pretty active in gradually and lightly penetrating into the interior, setting up scattered farmsteads with more or less friendship with Natives, spreading aspects of Russian culture and technology to them. The Orthodox Church will, on paper, completely dominate with most Natives in direct contact with Russians being nominal converts, provided the hierarchy does not look too disapprovingly on syncretic elements carrying over substantial amounts of Native religion in some form. They'll be baptized anyway. And churches will be common and widespread, even in regions overwhelmingly still Native in population.
Disease will spread relatively lightly and slowly, a lot of the worst epidemics coming mainly from non-Russian outsiders, since Russian policy will be to bring in controlled numbers of settlers over a very long haul.
(We can forget about the Hudson's Bay route; the British will firmly prevent that. All the Russians will be coming in via Pacific coast ports--conceivably some might sail from western Russian ports through the Atlantic, the straits between South America and Antarctica and up the South American coast to Puget Sound, or even possibly around Africa, through the Indian Ocean, and to America that way. They could also sail or steam to Panama or the other isthmian passages from Caribbean to Pacific overland, and then take other ships north from there. The vast distances involved are offset by the difficulties of moving overland through Siberia. But they definitely will not travel through North America from the east overland, unless perhaps friendly US relations encourage immigration through the USA. In this era there are no railroads going over the Rockies though so I think not. Maybe in the latter half of the 19th century?)
If the numbers of Russian settlers and Russian formal defenses, including I suppose a substantial naval squadron based on VI*, combined with US geopolitical commitment to the Russian alliance (a weak reed, but I am supposing it does prevail here for a rosy scenario) establishes in the US mind that Russian borders are there to stay, relations across the borders could be quite good. The Russian territories might become a refuge for Native tribes from farther east or south fleeing Yankee settlement, which would make the Indians share some resentment, but also firm up the defense, if the Russians can figure out how to accommodate the refugees. I expect the Natives will be being gradually and more or less voluntarily Russified, and on the whole quite loyal to their protective power, and contributing to it. As they shift to become more agricultural they can perhaps make room for the Native immigrants who must also go over to more intensive farming to get by on limited territory.
Thus the overall population density could be considerably higher than you were guessing, depending on how devastating Eurasian disease epidemics are. The more Natives are killed by those though, the more resistant the survivors will be, particularly those born of mixed marriages with Russians.
I cannot then foresee an essentially Native American nation, for none of the Native groups are large enough to dominate the rest. The cultural binder of Russian culture and society, highly modified for the conditions, will be what unifies them and gives them identity. But I do believe there will be a strong presence of retained Native customs and attitudes, many of which will become generic and shared by more or less pure blooded Russian descendants as well. Demographically, the Natives will be a very large percentage of the ancestry, perhaps a solid majority, as much so as in say Mexico. They will be Christianized and Russified, but in a peculiarly American way.
Territorially, the Tsar will be able to enforce claims to OTL Alaska, Yukon, BC, Washington State (somewhat eroded in the south) and I suppose Idaho and parts of western Montana and Wyoming. The border with the USA in the east will be formalized early and run on some ridge line or other, probably the Continental Divide, to a southern veer westward that allows a US corridor north of Mexican claims to Oregon. The British may make aggressive moves but they would in this scenario lack a base on the Pacific anywhere nearby, and coming overland from northern Canada, they will meet rough terrain inhabited by fairly loyal Natives appreciative of being largely left alone and reinforced with Russian outposts, arms and training and cadres of actual troops, perhaps few regular Tsarist troops but more recruited from the Russified heartland on the Sound or from among northern coastal peoples, Alyeskians proper. British settlements west of the Great Lakes would be at some risk when relations with Russia are poor, meaning Britain must divert resources to guard them. I don't see the Russians planning any grand offensives eastward, but if things go very badly for the British in North America they might wind up occupying to the shores of Lake Superior and Hudson's Bay, if the USA gives them some diplomatic cover. Probably the Yankees will concur with British interests that fix the extreme eastern border on the Divide though; if Russia and the USA are allied against Britain the US is going to want to claim Great Plains land for itself after all. We might concede lands around the Arctic Ocean.
This is to repeat a very rosy scenario that assumes the USA will not determine to gobble up the region we call Washington and lower coastal BC and to hell with the Russian alliance. The Russian alliance is not all that valuable, though if the rosy scenario holds it gains strength from the US and Russian mutual interest to avoid having to tie up too much in defending that border from each other.
As someone who is rather appreciative of the Bolsheviks on the whole, I would be interested in scenarios whereby such a grand RNA would possibly go Red in a 1917 type situation. Most people who want an Alyeska assume it would of course be firmly White, and there are reasons to assume that, including of course proximity to a strongly anti-Red USA and Canada. Given the likelihood that RNA would be a particularly pleasant and prosperous part of the Empire as a whole (it may start out low population and bucolic and peripheral, but if it can get through the early 19th century unconquered, I think that it will snowball, with the alt-Washington/BC part of it growing to at least OTL densities and via proximity to US trade and investment (closely watched by Tsarist officials of course) comparably highly developed to OTL western American states and British provinces. It would be rich compared to comparable areas of heartland Russia, even before gold is discovered--and by the time gold is found, the population and infrastructure in place would enable the Tsarist officials to prevent any wildcat secessionism along the lines of Texas or California even if they do have to let in a bunch of gold-rushing prospectors. They might limit that considerably and develop controlled Tsarist run companies to systematically do the mining though.
Such a development would drastically butterfly Russian history and the late 19th and early 20th century; we might question whether there would be any Great War at all, or if we suppose deep currents in history demand such a bloody confrontation around this time, the pattern would be completely different. With Russia and Britain on the same side, it is a roll of the dice whether the USA sympathizes with that side, or opposes it--odds favor the former. In that case North America is secure from war (unless Mexico is a lot stronger and on the wrong side) and American Russia is a treasure house for European Russia, funneling not only RNA goods but items purchased from nominally neutral USA and BNA as well. Indeed Russia probably has a much stronger and more modernized fleet well positioned to dominate the Pacific and can guard transport routes across that ocean, while the Trans-Siberian Railway would have drive behind it to become operational, for some value anyway, earlier.
What drastic effects might such a vast province in America have on Russian internal development though?
Anyway I don't see it going Bolshevik, though I've imagined scenarios-rather far-fetched I admit--whereby it could. Perhaps the added weight it gives the Empire prevents the kind of collapse Tsar Nicholas II presided over OTL and there is no revolution as such. Or if there is one it is limited to Old World Russia. This large North American province or set of them would provide a lot more room for political emigres fleeing a Red-won Civil War that drives the Whites off the continent--if there are any Reds, and if the greater strength RNA represents does not enable the Whites to simply prevail. Given the higher prosperity and development of the American provinces, and the fact that many or most of its people are not ethnically Russian but rather more or less freely and voluntarily Russified, I would think that among the emigres, the more liberal and progressive ones would be favored over hard-line reactionaries. The latter may have a safe if somewhat grudging refuge, but will not rule though the exiled Tsar or a pretender to the succession may formally preside. It is also possible that even if the Whites are driven out of most of old World Russia, that the Alaskan base reinforced by the richer more populous southern provinces can retain considerable if not very desirable footholds in Asian territory such as Kamchatka or the far northeast of Siberia.
If we game it out honestly, odds are Russian history is completely different though. Perhaps there is a revolution, but of a less desperately "workers of the world unite you have nothing to lose but your chains" type--perhaps radically socialist but started by RNA people despite their relative prosperity, and it is Old World Russia that remains reactionary.
Other interesting if perhaps nasty directions to go in involve the USA and RNA or a successor state coming to hostile blows in the 20th or late 19th century.