Richard III is an interesting character because he initially seems nothing more than a Machiavellian villain. He wants power at all costs and is willing to lie, seduce, cheat, and murder to get the job done. However, the play suggests he may not have been born this way, despite what the other characters think.
Richard says that his villainy springs from being treated with prejudice due to his deformity. His own mother balked at the sight of his "misshapen trunk"—or so Richard claims. Seeing as Richard is a regular liar, he could be lying even to the audience when he says these things. Like with Lady Anne, he wants to seduce us into siding with him by charming us and vying for our pity.
However, some disagree with this. Actor Ian McKellen, who played Richard on stage and screen, suggests Richard only ever tells the truth to the audience, making us co-conspirators, which could be quite true. Richard has been mistreated, and now he wants to make the rest of the world squirm.
One of the last scenes in the play also suggests there might be more to Richard than we see. After being visited by the ghosts of his victims in act 5, scene 3, Richard stares at his own evil as though for the first time and is horrified by what he sees:
O coward conscience, how dost thou afflict me!
The lights burn blue. It is now dead midnight.
Cold fearful drops stand on my trembling flesh.
What do I fear? myself? there's none else by:
Richard loves Richard; that is, I am I.
Is there a murderer here? No. Yes, I am:
Then fly. What, from myself? Great reason why:
Lest I revenge. What, myself upon myself?
Alack. I love myself. Wherefore? for any good
That I myself have done unto myself?
O, no! alas, I rather hate myself
For hateful deeds committed by myself!
This monologue suggests Richard does have a conscience deep down, though he decides it is too late to bother repenting. He will fight to the bitter end.
Some don't buy this scene. Critic Harold Bloom thought it was pure bathos, a failed attempt at the sublime on Shakespeare's part. Others think this moment elevates Richard from being a simplistic villain, making his downfall more tragic because he is keenly aware of his own wickedness. His moral self-loathing permeates the play's last moments.
So, is Richard a rounded character? I would say yes, and it's probably due to more than just the values of the period. He may be evil, but his charm, self-doubt, and moral perspective make him one of Shakespeare's most interesting anti-heroes.