Get all cars and tracks
At the main menu (after you choose a language), press X, X, Square, Square, Triangle, Triangle, Circle, Circle.
Hang back
As with the earlier Psygnosis Destruction Derby games, the only way to win the Demolition Circuit events is to hang back in the pack and hammer away at other cars for maximum points before dashing back to the front on the last lap.
Power-ups
The car repair, bonus points, and invincibility power-ups are always located in the same spots around the track, so you should memorize their locations and make plans to scoop them up as your damage level and points total dictate. (Just make sure the AI cars don't get to them first).
Victory
Try and perfect the Death From Above maneuver, which consists of destroying another car by launching yourself onto its roof after a jump. It's extremely difficult to execute, but it is worth 500 points and can pretty much guarantee a victory.