Don't give your hero too impressive a pedigree. A realistic hero should not be the aristocratic scion of a noble lineage of benefactors of humanity. That doesn't mean your hero can't be rich or have talented and important ancestors, but things should be balanced. Give your protagonist too impressive a background and you will create an epic hero, not a realistic one.
Have your hero deal with everyday issues. Realistic heroes can grapple with great struggles, but they must deal with the things all of us deal with as well. Getting along with family, finding love, living up to ones full potential, trying to act morally in a complex world, grappling with mortality, and searching for a community are some things that everyone can relate to.
Make your hero ambivalent. Epic heroes tend to primarily struggle with outside forces, whereas realistic heroes are more internally directed. A realistic hero won't fall in love happily ever after or become phenomenally successful without sacrificing aspects of his personal life. This type of hero will feel the mixed feelings and face the tradeoffs that everyone feels and faces.
Give your hero complex relationships. A realistic hero will have friendships that wax and wane, people who she doesn't quite understand, and even awkward and unpleasant folks she has to deal with repeatedly. Not everything is loyal companions and bitterest enemies.
Make your hero progressively evolve inside and out. As a realistic hero struggles with whatever tasks he undertakes; other things change as well. He will grow distant from some people who represent an old way of acting to him, and get closer to others. He will change the way he views and interacts with himself and the world around him. Tracing the ways these different changes intersect is one of the goals of a realistic story.
Tie the protagonist's evolution to his time and community. An epic hero can go out and fight monsters in a primordial forest, but a realistic hero is constrained by contemporary surroundings. Ask yourself if your hero is someone of her time, or is fighting against the currents of her time. Develop the way the heroes struggles relate to what her society is going through as a whole.
Start and end in the middle of things. A realistic story doesn't start with "once upon a time" and end in "happily ever after," and neither should a realistic hero. When the story begins, she is already involved in a variety of conflicts, allegiances, and undertakings, and when the story ends, there is still the next day and the day after that. Don't start things at too early a point or end them with too dramatic a finish.