When you unlock a new character, also purchase their character-specific power (for example, Howlin' Henry's Shuriken or Flamin' Raymond's Fireball) as soon as possible. This will let them to perform their special attacks as well as other characters to use them when you fill your special meter with power butterflies.

You only need the basic power upgrade to unlock it for all characters. After getting the first rank, focus on other traits like Attack and Defense if desired.