I was in Cartagena in late January and in the four days I was there it hardly blew. I guess that is fairly unusual for late January. It's a great town though and I would go again.
The kitesurfing beach is east of the city center, "La Playa Boquilla." It's a couple miles long, very wide and flat, and lined with high rises. It's not the blue water and white sand you'd get in the Caribbean islands.
I hung out at the AquaNautica kite school there. They've got instructors and gear, lockers to leave your kite overnight, and a roof deck with beer. Jesus runs it:
http://www.kitesurfcolombia.com/
+011-57 311 410 8883 (cell)
aquacartagena@yahoo.com
You can kitesurf anywhere on the beach (there are plenty of swimmers who are oblivious so that's a hazard if you're a beginner) and for convenience renting an apartment in one of the high rises would be the way to go. Cartagena is amazingly developed along the beach and a private rental is much cheaper than, say, a hotel.
I chose to stay in the old city though, just because I love the culture and I don't mind putting up with things like no hot water and street vendors. My small, practical hotel there was $80/night.
http://www.centrohotelcartagena.net/
The Bocagrande and Languito beaches are west of the old city and more built up. There is a small windsurfing school there that claims to have kitesurfing but the conditions are very sketchy. It could be a workable take-out for a downwinder if you were desperate for variety.
There are more locations an hour or two up the coast in the direction of Barranquilla. I understand there is a competition there this month in fact. Traveling in the countryside of Colombia can be dangerous. Cartagena is safe and I imagine the coast highway areas would be too.