Step 1 is to get a decent hammer. Pick one up in the game and pay the $$$ to fully repair it and bring it to level 4. If it's early in the game, the hammer will probably degrade quite a bit before you kill 15 zombies, so you'll want to stay pretty close to a workbench so you can repair it back up (or get a few spare hammers).
The technique to use is to always kick the zombies until they go down. Once they are on the ground, they take a few seconds to get back up and are easy pickings. This even works against groups of zombies. If you try to hit them straight away, you are likely to take a swipe before you get to 15.
This works throughout the game --> kick then kill. Makes the game much easier.
Also, try to hunt for zombies in a relatively open area so you don't get ambushed. The road leading from the lifeguard shack to the lighthouse is a good place for this. There will also be lots of munching zombies that are basically free kills. If you run up to them, you can usually dispatch them before they can stand up.
I haven't tested this, but I'd guess that getting a kill with another weapon resets your streak (my biggest problem was taking a swipe at kill #14). Your foot is really wimpy and it's unlikely that kicking a zombie will kill it so no need to worry about that.