This achievement is not hard, but it will take a lot of time to complete. The best way to do this is to play tons of battles and make your goal to damage Heroes. If you do this, you will complete this eventually. Battling against Heroes that can heal themselves is a good way to make this go faster.
Boosting your plants and zombies also helps. For example, if the opposing hero only has 5 health left and you have a plant or zombie with 6 attack or more, you will get that much damage, even if it is an overkill.
For plants, Mega-Grow is a very valuable class to use for this quest, as it has access to the most damage-boosting cards and tricks. Fertilize is a great card to use when it comes to buffing your plants. A good way to do this is by using Doubled Mint. If you manage to get a Doubled Mint with 300+ attack to hit your opponent, you will instantly complete the quest. However, cards such as Bananasaurus Rex, Potted Powerhouse, Pea Pod, Grow-Shroom, and Plant Food are also great choices, as they too can gain extra damage, either to themselves or to other plants.