I am going to put this simply because the solution that is currently posted didn't help me in the slightest. Like the achievement says there are 6 game modes to win, even though when you look at the in-game achievement it has 7 game modes (ignore guardian, that will pop when you do another game mode, that of which I can't remember).
The 6 modes you need to play are: warzone, execution, wingman, king of the hill, capture the leader and team deathmatch.
4 of these modes (Team deathmatch, warzone, execution and king of the hill) can be played by using the STANDARD lobby. In order to play Wingman and Capture the Leader you must access the FORCES OF NATURE lobby. It doesn't matter if you have the DLC or not, if you click on these modes without the DLC just keep pressing a to bypass the captions coming up.
You do not have to win these matches, nor do you have to play the entire thing. Just make sure that the in-game achievement down the bottom of the screen updates before quitting the lobby. Happy gaming!