One great way to make your characters sympathetic is to make them be lonely, abandoned, or rejected. Even a villain suffering from loneliness will evoke empathy (which can be a good thing, since villains should be both detestable and interesting to the reader).
Most people don’t like being alone and will sympathize with a character who is. Why else do you think bad boy loner types are so popular? Readers love Harry Potter partly because he’s being shamefully neglected by his aunt and uncle.
So make your character alone. Maybe they’ve pushed everyone away from them for some deep dark reason of their own. Maybe their personality makes them alone. Maybe they don’t know how to relate to people. Maybe they’re afraid.
Readers will often sympathize with someone who has been abandoned. This is a good way to make a villain sympathetic.
This is a great place to dig deep into your writer’s emotions and project your own fears, doubts, anger, and hurt into a character who has been left behind by someone they loved.
This is often tied to abandonment, but it’s a great tool to use to make your reader feel for your character. People love the underdog, the person everyone else thinks isn’t good enough. We love seeing that character fight to accomplish what others said he couldn’t.
So if you’re having problems with readers liking your character, add a little loneliness to their background or to the plot and see if that helps things along.