The chariot race is rather simple, just make your opponent's chariot take more damage then yours. Use the and to charge at your opponent, depending on which side he is on. Use the to protect yourself and release it at the right moment to flip over your opponent. Avoid the obstacles on the course, but try to push your opponent into them. And when you get the bonus boxes, press the in the direction indicated on the screen.
My best advice is to try to avoid the obstacles and try to push your opponent into them. That seems to be the easiest way to win.
This achievement can only be done in Olympic Mode.