You can pick your character by pressing Start at the main menu, then going to multiplayer then selecting your default COG character as either;
Minh Young Kim
or Tai Kaliso
Then all you have to do is play 10 Public online matches and hope you're on the Cog team.
If you want to do this quicker, play Warzone or Execution as they are the quickest game modes.
Note: When you join a game in progress it may not let you be your character for that game. Just finish that one and at the start of the next game you should then be playing with your chosen character.