I've heard of various ways to try and do this - but it's summed up very well here. So a big thank you to Jody Moller, who I'd never heard of but came across after a bit of googling on this topic.
1. The Redemption Factor
This applies to characters that start out unlikeable but over the course of the novel/movie grow, begin to really see themselves for the first time, recognise that they were unlikeable and change for the better. Think about Bill Murray’s character in Groundhog Day. At the beginning of the film weatherman Phil is painful. He is arrogant, rude and let’s face it sleazy. When he first starts reliving the same day over-and-over it brings out his worst characteristics as he tried to take advantage of every situation. But eventually – redemption and suddenly we the audience feel bad for him.
2. The Use of Humour
When we see a character behaving inappropriately but their behaviour is considered humorous then somehow we ignore the moral ambiguity and laugh – suddenly they are likeable, regardless of the way they act. Think of Melvin Udall in As Good As It Gets (portrayed brilliantly by Jack Nicholson). He is the definition of ‘unlikeable’ but he is funny and we love him.
3. The Badder Baddie
4. Morality Is All About Context
Give your Protag morals, they can be skewed morals, but morals nonetheless. Then make the Antag go in exact opposition to those morals – suddenly your protag is ‘likeable’. Often the ‘morals’ given to unlikeable characters revolve around protecting family, revenge etc… Think The Godfather.
Note that most of the examples given above fit into more than one category.
Like I say, whilst I've heard these ideas before, I've totally stolen the above from Jody's blog. She goes on to talk about Lisbeth Salander from the Millennium Trilogy by Stieg Larsson. So check out the full post here.