You can't do that with Remote Play. Remote Play allows the PS Vita to remotely access the PS3, but it doesn't give the PS3 any access to the Vita. There is no way to play a PS Vita game, a PSP game, or a Playstation Mobile game, on a PS3. If the game is a PS1 Classic or PS3-compatible Mini, then you can download it directly to the PS3 from the Playstation Store using the same PSN account that is used on the Vita. The Vita is not involved. You can't transfer any game from the Vita to a PS3 in such a way that it could be played on the PS3.
The primary use, in my experience, is that it allows you to use the PS3 while the HDTV is being used by someone else; much like the game pad that comes with the Nintendo WiiU. With enough network bandwidth it is possible to play PS3 content over the Internet while you are aware from home, but that is not always practical.