It really depends on the quality of the overall work. I love Graham Greene as an author – I’ve read eight of his novels – and I love his work despite the fact that all of his characters are prickly, weak-natured, despicable, or otherwise generally unsympathetic. I don’t think I’ve ever met a character of his whom I liked as a person. But Greene is a very shrewd chronicler of human behavior and spiritual yearning, and that’s what draws me to his work.
Likewise, Mad Men. I would pretty much hate to know any of those people, honestly (yes, even Peggy), but the show is so well-written in general (the structure alone of the episodes is remarkable) and the characters are believably constructed and very well-acted (yes, even January Jones) and the production design is impeccable, so that’s why I watch.
Then there are works that are flawed but have other draws, such as the unintentional hilarity of Matthew Lewis’ Gothic classic, The Monk. Everyone is heinous, but the novel is so over-the-top and lurid that it’s great fun to read. PG Wodehouse’s novels all pretty much have the same story, and Wooster’s an idiot and Jeeves a stuffy know-it-all, but the prose is so clever and laugh-out-loud funny that you have to keep reading. (Also, Wooster is an idiot, but a loveable one.)
On the other hand, if the general production (novel, tv show) is generally crap but there’s a character that I love, I might keep watching the show for a little while but that’s not really enough to hold me.
And if the production is crap and I don’t like anybody, then forget it.
As for what makes a character sympathetic, I ask that they resemble actual human beings and behave plausibly and consistently with the personality and history that we’ve seen. Does the work allow me to see into this character’s inner life and allow me to understand why he or she is behaving this way? If so, wonderful, you’ve succeeded. (This is why I love Dostoevsky so much – even his petty, shallow, venal, buffoonish, malicious characters are written in such a way that you understand why they are the way that they are. No one is a cartoon, everyone feels very real and life-like, so even if you don’t like them as human beings, you can’t help but appreciate them for being great characters.)