This can be obtained on x-box live, or on system link.
The "classes" are each individual person. Don't get that confused with thinking offense, defense, or support are the classes.
If you are doing this through x-box live, simply choose a character and play. When that game ends, choose a different character and play another game. I suggest starting with the scout and working your way right. That way it makes it easier to track who you still have to play with.
You can also get this achievement alone. Simply go to system link, and host a game. It doesn't matter if you are attacking, defending, or capturing points. (actually it doesn't even matter if you win or lose) Just pick a map that you can finish fairly quickly and put the round limit to 1. Then choose a character (-again i recommend the scout and working through the classes to the right-). Once you've played through each round with each class achievement should pop up.
this will also help in your Dynasty achievement. Because you are already well on your way to 20 wins by the time you've gone through each class.