Richard is malicious, power hungry and bitter about his physical deformity,
He is ruthless and plots to seize the throne by removing any and all impediments between him and the crown. Some critics have even surmised that he is suffering from bi-polar disorder (manic depression) – but perhaps that is taking the psychological evaluation a little far!
What sets him apart form other tragic heroes is the perverse joy he takes in his downfall, the way he stage manages his own tragedy and authors the kind of "sad tale" he imagines himself the hero of.
By definition, Richard is not a tragic hero.
The concept of a tragedy with a tragic hero was introduced by Aristotle. He said that a tragic hero was an inherently good man, who by some slip or mistake reached his downfall. This then creates sympathy within the audience, because we feel that we could have fallen prey to the same fate.
Richard is not an inherently good man. His personality itself is evil (although that can be defended by saying that he was shunned as a child, etc, it does not change that fact). There is no mistake that Richard makes to reach his fall. It is quite the opposite, because in fact, he is determined to be a villain.
However, the audience faces a great moral conflict. We pity Richard and sympathise with him to some extent. On the other hand, we know he deserves what he got because he is evil and evil is punished. The reactions that should be reserved for a true tragic hero, are here evoked for the villain. Hence, Richard can correctly be called a villain hero, but not a tragic hero.