Tsargrad comes from the translation of one of the Greek names of the city, which meant 'King city'

Russia never really had a claim for that city. Constantinople was raided by the Russians (they weren't actually Russians, but I will use that word) in the early 10th century, however they didn't conquer it, just got political benefits (such as trade without tariffs) instead of that.

Russia almost conquered Constantinople again during the Ruso-Turkish war of 1877-1878, but couldn't get the most out of the peace treaty.

So technically speaking, it was mainly due to geographical reasons, controlling the straits would give the country a massive boost.