When you find Cortex on Mystery Island there's an owl shaped lamp behind him. Hit the body-slam switch and it will rise up. Enter the cave, and go right. Have a chat with thethe drone at the top of the island and shell out 2000 coins to get the costume.

On the title menu, hold L1 + R1 and input SQUARE, CIRCLE, SQUARE, CIRCLE. (Unverified)

You must get all the Die-O-Ramas.