The way to unlock Oni difficulty

Arcade

Keep on going till to the Hard difficulty and keep hitting the rim (not the surface) many times until you get Oni difficulty.

PS2 and PSP versions

Keep on leveling up until you get Oni difficulty. (For CS4 and above in PS2)
Play in Hard difficulty only on Arcade mode. (For CS3 and below in PS2)
Get 25 crowns on Hard difficulty. (For "Taiko Drum Master" in PS2)

Wii and DS versions

Get 20 crowns on Hard difficulty.

iOS version

Get many crowns on Hard difficulty.