Smart ass answer: "when they got onto the ships"

Real answer: They always knew; expansion into North America was a conscious effort. the Bering Strait may be a relatively small body of water, though rough at times, but it was very obviously part of the Pacific. Considering the Russians expanded into North America in the 18th century, and European maps of the period were fairly well detailed - even that far north - the Russians definitely knew they were in North America. In fact, they counted on it; the prestige of having an overseas colony at the time can not be overstated; Russian pride hinged on the ability to beat the Europeans at their own game - something started by Peter the Great when he copied city planning from the English and ship building from the Dutch. Moreover, the wealth of furs brought in by the Alaska colony was pretty lucrative and so the expedition was definitely worth it.

If you're interested in the wider breadth of Imperial Russian history, I'd recommend Major Problems in Imperial Russian History edited by James Cracraft (1994). Though 20 years old, the book has a lot of primary sources - letters, directives, decrees, etc. from the time period - as well as essays by various experts which explain Russia's position in the world pretty well.