There are 4 Zombie classes that you will need to level to 5 in order get this Trophy: Foot Soldier, Engineer, Scientist and All-Star. You level these classes by completing challenges. The challenges are generally simple tasks like"Vanquish 5 Plants with X Attach" or "Revive 3 Allies", etc. Each completed challenge gives you a Star for that class and accumulating Stars will eventually get you to level 5.
If you run across a challenge that gives you a problem you can use a "Skip" to auto-complete the challenge. I used this skip function many times on my Engineer (wasn't a big fan of the grenade launcher). You get the Skip consumables from the various packs in the store.