One question that comes up in workshops sometimes is whether a protagonist needs to be likable. My usual answer is, no, he or she just needs to be really interesting. However, after watching "The American" and "The Town" during a long flight, I thought it might be useful to take that answer one step further.
The distinction I want to make is that "really interesting" is enough if you're not trying to win sympathy for your protagonist. If you do want that, you'd better put the protagonist up against people who are even worse. For example, in "The Godfather," while you don't exactly consider the godfather a great person, at several points he's in conflict with people who are even worse. But if you have your protagonist hurting or killing innocent people and you still want the audience or reader to feel sympathy for them...well, good luck with that.
SPOILER ALERT: If you haven't seen these movies and you plan to, then take into consideration that I'm going to reveal a key plot point for each of these films.
First, "The American." In some ways quite an interesting film, if slow. But within the first five minutes, George Clooney has shot his women friend in the head because she witnessed him killing two men who were after him. OK, that made him interesting but I think by the end I was supposed to be rooting for him. Nope, couldn't get over that shot in the head thing.
Second, "The Town." Well-directed and acted, it had the familiar "I'm doing one more job then giving up this life of crime and settling down with the woman I love" plot.
That works, except in this case he does that once...and then does it again--supposedly in order to protect the women he loves from a rotten guy. But I didn't buy that he couldn't go away with her at that point, so the last job--in which he and his gang are shooting submachine guns wildly at the police and innocent bystanders--wasn't really necessary. And we end with him giving the money to the woman with a nice voice over/note to her saying he know he has to atone for his past sins but maybe someday they can be together. Naaah, didn't buy that one, either.
Bottom line: if you have nasty protagonists, you really have to think through what you want the audience to feel about them and make sure you've included the right elements for that to work.
(Lots of good writing guidance can be found in my book, "Your Writing Coach," published by Nicholas Brealey and available from Amazon and other online and offline book sellers.)