Zombies aren't pretty, so face painting a zombie design is a chance to use up any dirty, muddy face paint colors you've got. Start by painting the entire face an ugly green, something putrid and unhealthy.
If you're going to be painting a lot of zombie faces, it's worth mixing up a bath of green by mixing either a little black, brown, or purple into a green. Add a little white if you think the result is too dark. If you're painting only a few, keep your paints clean by first dabbing a little black (or brown or purple) onto a few areas of the face, then apply some green, and blend the colors together. Don't worry if your color isn't completely mixed as zombies aren't known for their smooth skins, and some blotches are in fact desirable.