I am particularly interested in the 19th Century, as both of these nations expanded as it went on and both ended (with more or less success) their systems of slavery.
You might want to check out Saul Norman's Distant Friends: The United States and Russia, 1763-1867. Suffice it to say this isn't the first time that question's been asked and there's a reasonable number of articles on it. since it's actually an interesting thought. The two, for the most part, were rather friendly since neither was really in the other's way and the Western European Empires were greater threats to both countries. Remember they sold Alaska simply so the British would be less inclined to take it in the event of a conflict. (Like, say, the Crimean War)