How do I write a tragic antihero?

A good start would be reading some good books with tragic antihero protaganists. A short list would include: Wuthering Heights, A Clockwork Orange, Blood Meridian, and The Talented Mr Ripley. There are many more and they can be found in every literary genre.

Look at the flaws that make these characters antiheroes. The unheroic trait could be selfishness, pride, stubbornness, or any number of other things. Give some thought to how this trait could lead the character to a tragic end.