While this might go against the spirit of the game, try to keep a clean line through the race. Don't touch other cars and avoid any large objects, as they may cause you to spin. At the end of a race, the results screen will show the best lap time for everyone. If you are the fastest, that's one down, 49 to go.
If you're looking to get this done quickly, enter the "Extras" section of the menu and type WOTKINS in the cheat code area. This will unlock the Flatmobile, which is the fastest car in the game. Choose Figure of Eight 1 in Bonus Events, and change it to one lap. You won't have any trouble winning and thus receiving the fastest lap. Repeat this as many times as necessary.