Snowball fights are played in LOCAL multiplayer. You need a second controller to play snowball matches. For both your primary profile and your second controller, choose a character. Each round lasts 2 minutes, and it's best of 3, so if you lose 2 or win 2, you're done after 2 rounds. You can make one combination and then let the round finish. This way you'll still win. Assuming it takes 5 minutes per round, this achievement takes just over 4 hours.
Note: you also need 10 wins with different characters, so if you plan on completing the entire title, first go for that achievement.