While conversion was certainly welcomed, they didn't push the matter in Alaska (this is actually pretty on-brand for the Russian Empire for most of its history, actually)--Alaska was a commercial colony and it depended on native know-how and labor for fur trapping, and the last thing anyone wanted was to do anything that might upset them beyond what (from the Russians' perspective) was absolutely necessary to get the job done.

The biggest effect, ironically, came after the Alaska purchase. The ROC retained a presence in Alaska to serve the small Orthodox population that remained after the sale, and as Sergei Kan points out in Memory Eternal: Tlingit Culture and Russian Orthodox Christianity Through Two Centuries it was during this period that the Tlingit in particular began converting. As it turns out, a great way to get the Protestant missionaries coming up from the US to leave you alone is to be able to say "I'm already a Christian!"--and unlike the Protestants, the Orthodox Church really wasn't terribly interested in "Westernizing" the Tlingit so converting to Orthodoxy was an altogether sensible compromise that didn't require the Tlingit to change too much.