Makeup wise a good zombie needs to look organic
It shouldn't look "made up"
It skin should be leathery and dirty, like it just climbed out of a grave or something. Real dirt and moss grass cobwebs should be employed as much as possible

wounds can look fresh for effect but remember that zombies are the reanimated dead so an even more effective look is body cavity exposure. Any way to show depth will help

teeth need to be distressed as well as hair matted and mudded. Hair gel can be employed with "zombie dust" mixed in
Contacts are expensive but are effective (and dangerous)