Very limited.

Russian activity in North America was intended as a profit-seeking enterprise, not a land grab.

Starting in 1799, Russian America came under the monopoly of the Russian-American Company, which was focused on the fur trade -- then seen as the most profitable way to make money in northwestern America.

Fur trading didn't require large numbers of settlers or much infrastructure. New Archangel (today's Sitka) and Kodiak were the two principal hubs, but beyond that, activity was largely confined to traveling traders, missionaries, and scientist/explorers.

There were exceptions to the rule. Fort Ross, in California, was established to grow crops for the settlements in modern Alaska. Ninilchik, on the Kenai Peninsula, had a similar purpose.

Russia never had an issue with land, which was one of the driving forces behind the British settlement of the United States. Instead, Russian activity in North America was driven by the search for profit. That meant there was an incentive to keep the number of Russians in America low -- the fewer Russians, the fewer salaries the Russian-American Company had to pay.

For additional reading, I suggest Lydia Black's Russians in Alaska and Kodiak Kreol by Gwenn Miller. I've also written on this topic elsewhere in the subreddit. Search for my username, and you'll find my posts.