After Finny's accident, Gene's purpose was "to become a part of Phineas." What does he mean? Does he accomplish this?

Yes, this is the creepy part. Gene wants to be assimilated into his friend Phineas. Part of this is guilt over the accident and part of this has to do with some underlying homo-erotic feelings that remain unfulfilled or not quite understood.

Really Gene can't accomplish this, it's impossible to inhabit someone else, the best he can hope for is a "separate peace."