I'm making one (link in my signature), and I've a much developed outline, but I'm struggling to write since I've had to chose a chronological progression to keep the TL coherent. I have researched and I'm still researching, the subject, but I can give an outline :
1867 - 1897 : After a failed purchase, the Russian American Company monopoly on the land is abolished on the initiative of Finance Minister von Reutern (following the example of the Hudson Bay company though the premises of such an idea were already in the Golovin Report). The consequence is an increase of investment in the colony and in the development of its ressources, all of which relieve the burden off of the Russian state and potentially turn Alaska into a profitable venture. However, as with the United States IOTL, the Russian government is not very present, its role being mainly taken by the few military units, mostly Navy, and the Orthodox clergy, though a slight growth of government presence if felt from the 1880's. For administrative convenience, Chukotka and Kamchatka are eventually regrouped with Alaska as krais of a new governorate-general split from Far East.
1898 - 1917 : The discovery of gold in Klondike and then in Nome causes the government to take a more proactive policy. The troubles brought by the gold rush and the overwhelmed local authorities lead to Russia significantly bolstering its garrison and overhauling the government apparatus (by local standards I mean). It's at this time that the first railroads are built, like iotl, on foreign capital (one for transit into Yukon, the other to export of Kennecott copper), and that the Russian government sponsors some colonization from western Russia (not big numbers, but important regarding the colony population). The border is definitely fixed under US mediation. During WWI, though conditions of life are rather harsh, Alaska's proximity to West Coast and Canadian ports make it less suffering than Russia proper.
1917 - 1929 : The Russian revolution leads to some disturbances, but minor and unconsequential ones in regard of what happens in Russia. General Kuropatkin, thrown out of Turkestan by Bolsheviks, is sent to Alaska by Kerensky. He governs here when happens by the Bolshevik coup and refuse to aknowledge the new power. He has not much difficulties to resist since Bolsheviks have a hard time in Siberia after the dissolution of the Constituent Assembly and the Czechoslovak Revolt. It's only after the collapse of Kolchak and the fall of Crimea that Kuropatkin becomes de facto the leader of White Russia. Because of Bolshevik focus in Transbaikal and the Maritime Province and elsewhere against Crimea, Poland, Tambov rebels and Krondstadt mutiners, Kuropatkin holds onto Kamchatka and contest them the control of Yakutia until mid 1923. A last stand has been prepared at Okhotsk (the only possible way to Kamchatka which geography makes impossible to bypass) but the city eventually falls after one and a half year siege in late 1924 at the hands of either Frunze or Tukhachevsky. In the meantime, Kuropatkin has died, and Alexander Guchkov, his unofficial prime minister, has succeeded him as head of the White Russian government. The fall of Okhotsk and financial strains make holding onto Kamchtka impossible, and Petropavlosk is totally evacuated during winter 1924-1925. Whites still hold onto the Komondorski Islands, Wrangell Island too although presence is either occasional or symbolical (Soviets have not yet thought of taking it but that wouldn't be a great difficulty), and theoretically North Sakhalin though, in the facts, has become a Japanese province in all but the name (being occupied and administered by Japan with the Russian governor a powerless figurehead).
While the civil war rages, Russian vodka king Vladimir Smirnov builds an empire on the West Coast thanks to the prohibition and he even secures immunity from the US government by playing the middleman to funel money and weapons into White war effort, though he is smart enough to hire local bosses as intermediaries to keep his hands clean. Smirnov would eventually come to weigh large political influence on local politics and culture. To diversify and reinvest his wealth, he would involve in filmmaking business and would push for slod deg racing and bandy in olympic competition, making the latter one of the most popular sports in northwestern america (ittl, ice hockey is more an eastern thing).
Though large numbers of people have taken the road of exile, not many have elected to settle in Alaska, though the fraction who does it has a relevant impact of demographics and culture. Aside of the civil war, food supply is an important issue and later in the decade, some efforts are made to develop agriculture in the Matanuska-Sutsina valley, and also in the upper Tanana valley. The construction begins on a railroad in the Matanuska-Sutsina valley to stimulate local agriculture and of another to extend the Kennecott Railroad to the Tanana valley, and also on new roads, but limited funds slow down the progression, and it wouldn't be until the Pacific War that they would be truly completed and operational.
Politically, I've not yet fixed the details, but Guchkov would not continue to govern Alaska until 1929, possibly replaced by Milyukov or another politician. Administratively, the government's apparatus has partially collapsed and its authority is often ignored, though not outright contested, and due to the absence of legal cadre (no constitution to speak of), rule is made by decree, and this would be a constant until 1964.
1930 - 1937 : Since Alaska has been heavily reliant on foreign help, it is very badly hit by the crisis of 1929 and the government gets eventually overthrown by the military and replaced with a junta headed by general Mikhail Diterikhs. Until Diterikhs' death in 1937, military rule translates into heavy handed repression as part of a witch hunt organized against alledged Soviet 5th column; relatively quiet up to this point, Alaskan Natives are victims of it, being displaced. Under this regime, a number of gulags in the Arctic, no less harsh than their Soviet counterparts. The repression goes along a reshuffling of the administration that is completely taken over by the military; this move puts an end to the administrative chaos which existed since the revolutions of 1917.
Diplomatically, the regime is quiet friendly with fascists and nazis but has abstained from offending its American protectors, but after FDR election and his rapproachment attempt with the USSR, the junta has drifted more towards Japan. Japan may take advantage of the situation to perform some exploration of oil resources but wouldn't find anything substantial.
1938 - 1964 : Diterikhs' successor, for the sin of being too close to pro-Japan, and germanophile, faction, is overthrown in a coup instigated by the United States (part of the motivation is backlash against Japanese invasion of China and atrocities they committed here). In 1941, a coup attempt instigated by Japan fails but announces a Japanese invasion of the Aleutian Islands (and possibly also the Cook Inlet) after the attack of Pearl Harbor, before being expelled by late 1943. During the Pacific War, Alaska experiences an economic boom because of US military presence, and the construction by them of important infrastructures such as roads, ports and airfields. This presence, though threatened a time by the end of the war, would be maintained because of the Cold War. Besides that, repression from Diterikhs years has much ceased, but at the same time, corruption spread in the ranks of the junta as more and more US dollars pour into Alaska, taking a dramatic turn in the 1950's when the regime comes to collude with organized crime. Indeed, after the war, increasing tourism increases, lax fiscal laws (inherited as a courtesy of Vladimir Smirnov's money laundering operations begun in the 1920's) and legalized gambling have drawn in the mafia, and though their activities are not as important as in Cuba, the comparison would be made (disparitions ie kidnapping, torture and murder of activists the Okhrana would become a frequent occurence into the late 50's). On the institutional field, the abandonment of the Diterikhs policies and the participation of Alaskan Russian military in the Pacific War, along the ageing officers corps (not much renewed since WWI), has made the military administration redundant and anachronistic, leading to the development of a significant civil service. This last reform has in itself the germs of a generation renewal, the birth of an Alaskan identity separate from Russia, since it will increasingly revolve around people born or raised in Alaska for whom the civil war was only a dark childhood memory. As mentionned, this goes into the 1950's along an increasingly vocal contestation, as much political (especially by Natives) as cultural, due to counterculture (Rock 'n' Roll) brought in by US soldiers.
1964 : The Good Friday Earthquake hits and ravages southcentral Alaska, home to most of Alaska's population. The inadequacy of relief efforts are the "straw that breaks the camel's back". Popular discontent against the regime erupts open. In Yakornaya, when the local garrison mutinies after comes an order to suppress the contestation, it becomes a revolution. Other mutinies happen and the junta officials are either arrested or flee the country. American reaction is all along expected and feared since the Cuban precedent is still fresh in the memories, but invasion is averted when the provisional government, barely created, draws the 83 years old Alexander Kerensky out of retirement to make him president as a guarantee for the United States (noone is going to accuse the very person Communists overthrew in Russia to establish their regime of collusion with Moscow).
A constitution is written, and to the contrary of previous works under the junta but kept in the boxes, this one ignores mainland Russia and is tailored for the government of "the krai of Alaska". Though not explicit, this is de facto a renunciation to claims of recovering Russia, but while in the minds of the writers, that has been essentially a practical measure, a necessary aknowledgment of the fait accompli, there is also those, among the youngest generation, who think of this being a step more forward declaring outright independence.
In this new republic, that takes some of the features of Mexican system, the President is elected for a single 6 years term, along a Vice President who isn't term limited. The legislative body is made of a Duma renewed every two years; due to practical reasons (because of large underpopulated areas), the election is made through proportional representation with party list.
In case the presidency becomes vacant, early presidential election is set to coincide with the next legislative election, with the vice president being only acting president but able to run.
1965 - early 1980's : The discovery of oil at Prudhoe Bay still happens on the same schedule, in 1968. The promise of oil revenues and sold concessions, allows Alaskan government to push for infrastructure projects such as the Transalaska Pipeline necessary to bring oil from Prudhoe Bay into southern ports for exports and bring oil taxes into the treasury, and the Rampart Dam to address the issue of costly energy supply and provides for additional revenue through export and developing aluminium industry. Though initial work, mostly survey, planning and preparation of logistical infrastructure, could be done, it's not until the oil crisis of 1973 that Alaska manages to secure enough funding for the projects to go at full speed with low interest loans and subsidies provided by American government and banks. On the cultural field, the construction of the dam and the development of oil resources have led to rise of influent environmentalist and Native rights activists. To settle the matter and allow the works to goes on, a compromise has been agreed: an electoral reform to increase the weight of Native electorate, several bills to define and extend native rights, protection of their culture and lands, and also creation of large protected areas of wilderness, not counting a tightening of regulations on oil drilling and exploitation.
The government also sets up state run welfare programs similar to the Nordic Model. Though oil wouldn't go into export before 1977, Alaska's foreign debt begins to grow at a rate that isn't sustainable in the long run. In the short run, the revenue from oil taxes after 1977 and the oil crisis of 1979 has limited the damages, but the speculative bubble that has come out of this situation threatens to doom this Alaskan Miracle.