Honestly, I don't think that Alaska wouldn't be able to remain Russian for long. By the time of the Alaskan Purchase, Alaska was of barely any use to Russia: transportation and logistics are extremely difficult, the weather harsh and the fur trade long been declining. I also doubt that the Tsars would ever mind about investing in and developing Alaska. Once the Gold Rush happens, Alaska is going to receive an influx of American and Canadian miners, which would be an ideal pretext for annexation by the US or Canada.
However, with the right PODs, we could have Alaska ending up as a Russian Quebec with a plurality/majority of the population being settlers from the Russian Empire. Alaska's cold climate and lack of arable land wouldn't be that appealing to immigrants (especially those from European Russia or Ukraine), but AFAIK the conditions there are probably not that different from Siberia.