Play and complete a game as every class at least once.
Please note, this does NOT mean you have to play from start to finish without switching class. I joined a game late on the last round of Dustbowl, I chose Scout, and when that final round finished, I unlocked the achievement. This shows you don't have to play the entire game, just from when you join, til the game ends.
If this was a glitch and you do actually have to play the whole game from start to finish please let me know, but I believe you don't have to.