Although the achievement description states 'Quick Races', this does not need to be done in the 'Quick Match' public multiplayer mode.
You can boost this quickly and easily using a private lobby with a friend (or a second console and account if you have one).
Simply create a private lobby, invite your friend/alt account and then simply win five races.
The other player(s) in the lobby do not need to finish the race, or do anything. They can simply idle on the starting line until the race automatically ends 30 seconds after the winner crosses the finish line.
This will also count towards the following achievements: