I think it would be fairly important to me. It's the characters like Harry Dresden or Jack Reacher that usually get me hooked on the series. What would really grate on my nerves and turn me off a book would be the Mary Sues....
so yeah, how believe-able the characters are would be important to me just as the importance of being able to connect/relate to them!
For me, the story is also just as important as the characters. I do read Clive Cussler now and then (whether it's about the adventures of Dirk Pitt or Kurt Austin or Juan Cabrillo), and I don't even mind if the story is from the POV of a "villian" (like Dexter) if the premise is interesting enough. It's just not going to be the same if the entire book is about Jack Reacher doing nothing but telling you how he's learning to flip pancakes! (oh wait, there's Odd Thomas, and I like him too, but then again he's not just flipping pancakes, but I digress lmao
).