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 L + R and input BLUE, RED, BLUE, RED. (Unverified)

On the START GAME menu, hold L + R and input GREEN, BLUE, YELLOW, RED to disable the HUD. (Unverified)