You should get this after finishing the final Hero mission, just make sure that during your boss fights, you make the killing blow on each of these chracters while controlling Batman. If you kill them while controlling Robin, the achievement will not unlock. Replay the necessary story missions in Story Mode if you messed up.
- Joker: To the Top of the Tower (Episode 3, Chapter 5)
- Two-Face: The Face-Off (Episode 1, Chapter 5)
- Riddler: The Face-Off (Episode 1, Chapter 5)
- Catwoman: There She Goes Again (Episode 2, Chapter 1)