The main obstance is that no matter how enthousiastic Russia gets about colonising its North American possessions, it can't build a Trans-Siberian railroad by magic: even if it gets going earlier, it won't be there before the 1830s or so, I'd say. And that's a very optimistic guess. Before such a route, moving a lot of Russians to the American North-West will be tricky. Whichever direction you try to approach the region from, you have to cross a lot of land and at least one ocean to get there from the Russian heartland.
By the time the Russians can really start colonising, the British and the Americans will be so heavily encroaching on the region that winning will be difficult. holding on to Alaska is possible. Enlarging it can be managed. With a lot of luck, most of the land west of
Rupert's Land could end up being Russian-owned. Yet Russian-owned isn't Russian-populated, and Anglo srttlers from the east may simply "pull a Texas" and take over most of the region anyway. Besides that, even if Russia ends up with more of the North-Western regions of North America, neither Britain nor the USA will tolerate Russia holding the Seattle Bay area. So the southernmost reaches of Russian North America are almost certainly going to be north of there. This denies Russia a good port, which will cripple Russian North America.
So no matter what, you end up with "super-sized Alaska" at best. (okay, that is somewhat uncharitable. It's a pretty cool region, and even if sparsely populated, the natural resources are great to have.)
However... if you want something more ambitious, you need more than "Russia gets busy colonising at an earlier stage". Your best bet is some kind of scenario wherein Britain and france fight for longer over (the Eastern parts of) North America, ideally culminating in French victory. The french had fewer settlers available, which means they'll likely be later to start encroaching on Russian America. Even better would be to have Spain go to major trouble as well, somehow, leading its colonial empire to be much weaker. This will cause Spain to pull resources away from the periphery. All of these factors would leave Russia free to annex strategic places, possibly as far south as the San Francisco Bay area. Now
that would be a very powerful Russian America!