Press the following buttons on the main menu: X, Square, Circle, Circle, Circle, 
Triangle, Square, X. 

Boost:
Defeat Boost during story mode.

Chick:
Defeat Chick during story mode.

DJ:
Defeat DJ during story mode.

Doc Hudson:
With the Gran Prix Cup 1

Sheriff:
Win the Gran Prix Cup 2.

Snot Rod:
Defeat Snot Rod during story mode.

Wingo:
Defeat Wingo during story mode.