I purchased Ultimate Marvel vs Capcom 3 from the PSN for the PS Vita. I've read that I can access DLC stuff for this game from either the Vita or my PS3. What I was wondering is, can I also transfer the entire game to be played on my PS3 or do I need to buy a separate copy of the game itself for the PS3?
No. These are two different games and require separate purchases.
While some cross platform PSN games give you versions of the game for both platforms when you purchase them (for example, MotorStorm RC and the future release of When Vikings Attack), this is not the case for UMvC3.