To access the cheat mode press Up, Down, Down, Up, R2 at the Wing Commander Copyright Screen. If you entered it in correctly you should be brought to a level select screen.
To use the kill cheat in battle press L1+L2+Square and watch the enemy blow up. Caution: you can also blow up the good guys!
Thanks to Revolution readers Airbond, Michael Patten, and The Maniac!