The Zombie are notoriously difficult to kill, the base reason being because they are already dead. In order to kill a zombie, then, you'll need to do some extensive damage to the thing that's keeping them alive. That thing is the brain. There are multiple ways to kill zombies, of course, and you can be as creative as you like.
Remember that each of these methods requires that you destroy the brain, so make sure you focus most of your energy on the head. Destroying the other organs of a zombie will do absolutely nothing - it'll just direct its attention more toward you.