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!