The Puppies and Kittens achievement in State of Decay 2 is pretty self-explanatory. You need to kill 100 zombies. This will come naturally pretty quickly but if you really want to expedite the process, get in a car. Driving around and hitting every zombie on the way, especially hordes, will most efficiently boost your kill count.
You can also stop for infestations and plague hearts you see on the way, as these both spawn plenty of zombies as well.