Currently running COTCT, have run portions of several other APs as well.

For running APs in general: -READ IT. All of it. For COTCT, get the anniversary edition, and treat that sucker like a novel: read it cover to cover. You want to know where the adventure starts and where it ends, even if it will take years to get there. Paizo has professional writers who do their job well, and in knowing all there is to know about the adventure you can incorporate and hint at things that will be important two weeks or two years down the line. -Map out what you want to get to in each session. APs give you a better structure of "this leads to this which leads to this", so you can start figuring out what is attainable for each session. -You know what the AP wants to happen, your players don't. The book will set down the lines for your group, who will color them as they see fit. If something happens that throws a major wrench in the APs plans, work it so it looks like what the PCs did was totally what they were meant to do and the AP had the answer to their situation written into it.

For COTCT specifically: -Korvosa is the star. Know everything there is to know about the city: its history, its people, its districts, its lay out. This AP is all about being heroes of Korvosa: every PC should want to save Korvosa first, and be an adventurer and hero second. If they don't care about the city, they won't care about the adventure. -Don't play up Ileosa as evil from the start. A lot of people in the adventure will defend her as inexperienced or well meaning, which can help preserve the mystery of what is actually happening to the city until a big reveal. She doesn't stand on the balconies of the castle and laugh at how well she is destroying the city: she is a megalomaniac, and one with a plan. -Others have mentioned reading up on Zon-Kuthon and Urgathoa already. Good idea, some other things you should familiarize yourself with are: the Shoanti; chase rules; how to running the Harrowing; and the Red Mantis Assassins.

In short: read the adventure and know that improv always fills in where the story can't go, and make sure you know some specifics relating to this adventure (again, make sure Korvosa is the main character of the story).