Out of the ten multiplayer trophies, five of them relate to ranking up each class, Assault, Medic, Scout, and Support. To get all of them, you need to get each only to Rank 2 (which is pretty generous). There's a gold trophy for reaching Rank 2 with each class and a silver trophy for getting to at least Rank 1 with all classes (which will come naturally if you are aiming for all the Rank 2 trophies).

You rank a class exactly how you'd rank up normally, by completing objectives, getting kills, etc. The only difference is you need to be playing as the class to rank it up. If should take you about 4 hours or so to get to Rank 2 with a single class, but it ultimately just depends on how good you are at the game.