There are 4 Plant classes that you will need to level to 5 in order get this Trophy: Pea Shooter, Sunflower, Chomper and Cactus. You level these classes by completing challenges. The challenges are generally simple tasks like"Vanquish 5 Zombies 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 Cactus (I'm not much of a sniper). You get the Skip consumables from the various packs in the store.
For the Plants side many of the challenges can be completed in Garden Ops which is typically easier than in the Multiplayer.