It may seem odd and shallow, but I find I can connect more with a character if they remind me of someone I know in real life, either personality-wise or through some parallel struggle/event. I start to picture the character as that person and tend to completely disregard the physical description the author provides (even if there's a picture of the character on the front cover) in favor of something real that I can recall easily. If the character doesn't match up with any personality I know, then he or she becomes bland, even if they are the most well-developed and engaging character to ever be created. For example, if there's a female character that I immediately connect to my hypothetical real-life friend "Stephanie," while I read the story I find myself comparing the character's actions to see if they match up with Stephanie ("She is just like that!" or "She would never do that!").
This is a hard question, because why we are drawn to certain characters might not be completely revealed to us. Partly why we are drawn to this character and not that one, might forever remain hidden for us. I love Christopher Snow, from Dean Koontz, Fear Nothing, and Seize the Night. I believe the reason is is that he is a weirdo. I like weird eccentric characters. I like different. This is probably why I like vampires such as Lestat. He is another one of my favorites. I love Dexter, but I really didn't want to like him. Who wants to be sympathetic for a serial killer? Who wants to find themselves cheering for the killer? But at least he has a code and only kills sickos. I like quirky also. I also like intelligent characters. But they must be eccentric and intelligent, or different and intelligent. I think I like Tom Sawyer, because he is an odd ball. Odd Thomas, Toad in Lost Boys, Mina in Dracula, The weird friend in Fright Night, The MC in Fight Club, Brad Pitts Character in 12 Monkeys, and the list goes on.
For me, it's either a quirk in the personality or vulnerability - not all the time because I don't like characters without back bone, but just a brief fleeting glimpse that perhaps they're not as in control as they thought - and I'll keep reading the book to see if they crack - either hoping they don't or willing them to. Cruel, but true.
I'm proud to say that I was the friend at the rooftop bar when this conversation began. I think I like characters who are not like me and who I often love to hate. I think of Mr. Darcy and his arrogance that I almost couldn't believe was real, which come to find out later was not who he really was. I like those characters that I see slither away on the page, but you know the writer had them there for a reason, even if just for a minute, and took them away for a reason. It is these characters, not just the main/typical antagonist/protagonist situations, that keep me wanting more and help me fall deeper into the text. Keep writing CRP!