Where can I find the tavernkeep?
It also states that he can only be found after defeating the Eater of Worlds or the Eye of Cthulhu. I've already beaten both of them before the update, do I have to kill them again to spawn him?
You do not need to kill the Eater of Worlds or Brain of Cthulhu again to cause the Tavernkeep to spawn (he automatically appeared in several end-game worlds of mine after updating from 1.3.3 to 1.3.4). I believe the rules for his spawning are essentially similar to the Bound Wizard, Goblin Tinkerer, and Mechanic, in that he will spawn somewhere near your character as you travel around. As the comment above mentions, you can now locate him using the Lifeform Analyzer (introduced in the 220.127.116.11 patch).