So, you've decided to play as a zombie. Zombies are tricky to play since they are, as strange as it seems, outnumbered and easily turned to vampires. But you didn't come here to listen to someone preach about game balances, you came to find out how to play as a zombie without waking up as something else the next time you log in (well, that's not the ONLY thing...).
If starting a new Zombie character (initial setup) try the Festering Cadaver class, with Biohazard. With this skill, you can effectively combat your greatest threat, vampires. As of 11/05/07, the skill enables you to "turn" vampires you kill into zombies. Biting attacks have less of a chance to hit but do more damage. With Hunger Pain you increase hit and damage potential when low on AP and Grim Feast gives a 45% chance to heal 3 HP on a bite attack. Concentrate on developing bite attacks first for these two skills. You should then go on to improve your Hand attacks. They have a better chance to hit when all the hand skills are maxed (55% to hit with 3-5 damage), but do less damage. Still, hand attacks can prove to be effective when you don't have alot of AP to spare. Next would be Corpse Feeding as this will allow to you to regain HP.
Cellular Degeneration is nice because when revived let you choose to rise as a human or as a zombie, saving the time (and AP) to commit suicide to regain the zombie status; Adrenaline Boost gives you an extra HP bonus (+10HP) when a human use the serum on you.
The Solid Grip and Flailing Battery skills are not worth purchasing, as maxed hand attacks will have a higher hit rate than any weapon attack, but it still will allow for some good humiliation kills. (who could imagine getting beat to death by a zombie with a Nightstick?)
Non-attack skills to concentrate on first are Intellect and Identify Life. These two skills allow you to open doors and search for hidden humans.
Day to Day Strategy
Here's a simple plan for starting zombies that want to make kills, but also want to avoid becoming a vamp.
Starting at full AP, attack/kill a bunch of enemies you find in the open (it may take a while, but they are there), leaving maybe 10 AP or so (less if you know the area). Then, find an unbarricaded "building", and commit suicide from it. Yes, you just died, but unless a human comes and randomly revives you, you will still be a zombie when you get back up. Plus, it doesn't technically count as a death on your stats page. You will have 20 hp upon standing up, but seeing as you will be doing nothing but hit and run for a while, it's not a problem.
A nice and "easy" way to gain XP is destroying barricades. Look for a sledgehammer or a fire axe and take out every human effort to stop you from getting inside. It may look strange to hunt for something that isn't alive, but you'll recieve 1XP for each successfull hit and some of them can get 10 hits before tearing down; also a barricade means (not all the time) that there is fresh meat waiting for you inside.
Zombies have the fewest AP (40) of the three races. Stay near a target rich environment. Currently the Fort is the best hunting ground for Zombies.
General Survival Tips
Find a walkie talkie and join a group as soon as you can. Group members can help you find prey and protect you if you run out of AP. If you get revived try to find a Kevlar Vest and put it on. It has a chance to reduce damage, with the amount reduced increasing as the damage increases. If you do get revived make kills as a human to get XP and then commit suicide. Take your XP and buy another Zombie skill. If you get "vamped" just check the wiki for the nearest revive point and head over there (and maybe bite a few people on the way there, AP permitting, it's always nice to go back to zombie form with a huge load of XP). More preferrably, you can try to ask a group member or a fellow zombie to give you a bite.
Don't waste your XP on skills you don't want or don't need. Since every time you buy a skill, the next skill you buy increases in price, buying pointless skills could hinder you when switching to another race.