To unlock the extreme Match Race, beat all three difficulty levels and then once you have completed Avalanche Alley: No Way Out (the last level of the "Hard" difficulty), wait for the credits to end. An extreme course will now appear at the Match Race main screen.

On Match Race I would sugest using Kemen because he is the fastest, but he is hard to control so watch out for that but if you can control him you'll be good.