First, you have to get infected. This is easy to do as there are Conspicuous Crates all over -- I clicked on some from Booty Bay -- which will give you a 10 minute debuff called 'You're Infected!'. The only way to cure the debuff is to be in the vicinity of an Argent Healer [EDIT: It can be cured by Paladins, Priests, and Shamans, too. Those killjoys!], who will heal you automatically with several yards. If you want to serve the Lich King, stay away from the major cities until you get turned. If you don't get cured within 10 minutes, you will, like me, turn into a zombie!
[UPDATE: If you transform into a zombie, you will need to keep on killing things. Otherwise, you will constantly lose health until you die and return to your normal state. The ability Bite! is a passive effect that happens whenever you attack, so you'll constantly need to keep attacking. This prevents zombies from traveling very far without killing.]
When you turn into a zombie, you'll get a whole bunch of unique abilities. The first is Mangle!, which is your basic melee attack on a 3 second cooldown. It's fairly weak, but the best part of the ability is that it seems to shave minutes off infections. Attacking players or NPCs will infect them, so be ready to spread the plague. Players and NPCs that die with infections will transform into zombies immediately, as will players whose infections run the course naturally or through zombie attacks. With a little cooperation, you'll soon have a zombie army of your own!
You also have Retch! which is an AoE that slows down enemies... this is good because you move slow as, well, a zombie when you're transformed. It also heals your fellow zombies, so use it wisely and often! You can use Beckoning Groan! to attract other zombies to your aid -- the NPCs that you killed, specially -- and create undead unrest everywhere. There's Lurch!, which is kind of like a Dash or Sprint for zombies, which is great because moving slow gets really annoying. The cool thing about Lurch! is it dispels movement impairing effects, too. You'll need that when city guards tie you down. And finally there's Zombie Explosion! for when you want to end all that misery! It's a channeled spell that will make you explode! Kind of like a self-inflicted Corpse Explosion.
Here are some tips to create maximum mayhem as a zombie: while you become hostile to everyone else, you can still whisper members of your own faction and talk in various channels. If other players attack you, you can just whisper them -- you'll have to type out their names as the selection menu won't allow you to do so. Tell them to get infected to, and there's a good chance they'll join in on the chaos! Attack players and NPCs with the debuff, spamming Mangle!, which shortens the You're Infected timer by a full minute with what appears to be every other hit. Just remember that you won't have access to all your normal abilities as a zombie, and won't be able to mount up or hearth. Well, now that you've got that teaser, what're you waiting for? Go forth and zombify!