Okay, here's a process that will net you 4 achievements in a matter of a couple minutes.
1. Do a versus fight, but DO NOT quit afterwards.
2. When it gives you the "A: Next" prompt, press A to under the post-fight menu, click on "Save Replay" (this pops the 'A Fight To Remember' achievement).
3. From there go to the main menu and select Extras > Spectator > Fight Viewer, and pick the fight you just saved, and 'Replay' (this pops the 'On the Edge of Your Seat' achievement once the video starts)
4. While the video's playing, pause and select 'Take Photo'. When it returns to the playback, mess around all you like with the camera positioning, and press RB to take a picture (this pops the Say Cheese! achievement). Quit the playback, or let it finish if you want to watch more.
5. Finally, go to Extras > Album and you will pop the 'Ahhh, Memories' achievement.