There are two different ways to open up characters. You can either play through battle mode and gain characters as you win, or you can unlock them by earning that character in Story Mode. The Final Stage level can also be opened up by completeing Story Mode.

Unlockable	How to unlock
Blossom	Win 12 matches in Battle Mode or beat the first level of thePowerpuff Girls level in Story Mode.
Bubbles	Win 7 Matches in Battle Mode or complete the first mission of the Powerpuff Girls level in Story Mode.
Captain K'nuckles	Win 3 Matches in Battle Mode or complete the first 2 missions of the Flapjack level in Story Mode.
Captain Planet	Win 75 matches in Battle mode or complete the Story Mode.
Father	Win 42 matches in Battle Mode or complete Story Mode.
Final Level	Complete Story Mode.
Grim	Win 25 Matches in Battle Mode or beat the first level of the Grim Adventures in Story Mode.
Mojo Jojjo	Win 63 matches in Battle Mode or complete Story Mode.
Monkey	Win 18 matches in Battle Mode or complete the first level of Dexter's Lab in Story Mode.
Samurai Jack	Win 33 matches in Battle Mode or beat the first level of the Samurai Jack stage in Story Mode.
Vilgax	Win 52 matches in Battle Mode or complete Story Mode.