Each of the four classes has 10 ranks, and you can progress in rank by completing the associated challenges. All class challenges involve killing enemies a certain way, and none of them are any harder than the others so pick whichever class you prefer and stick with it.

Under the loadout menu you can see which class you are currently playing as, as well as the challenges for your current rank. These challenges can be pinned (by hovering over the class icon in the loadout menu and pressing ) to your interface, accessible by pressing  while in game. Make sure to pin your challenges, and check them frequently to make sure you're meeting the requirements to complete them. Upon reaching rank 10 with any class this trophy will unlock.