The play opens with a soliloquy by Richard, Duke of Gloucester. He tells himself--and his audience--that peace has settled on the land and everybody is happy except himself. He describes himself as being so misshapen that he cannot enjoy the pastimes that are associated with peace, including dancing and love-making. And then he expresses his wicked intentions in unmistakable terms:
And therefore, since I cannot prove a lover,
To entertain these fair well-spoken days,
I am determined to prove a villain
And hate the idle pleasures of these days.
So, according to Shakespeare, Richard decides to become a villain because of his deformity. Shakespeare probably did not believe that a villainous character should produce deformity. Edmund, in King Lear, for example, is an outstanding Shakespearean villain, but he is very attractive to both Goneril and Regan.
Richard's self-assessment can be read by referring to the link below. What is remarkable about him as a villain is that he can see himself with such unflinching objectivity. From the beginning he shows himself to be cold, rational, remorseless and pragmatic. He is a memorable villain who is fascinating in his utter malevolence.
You can consider this in terms of both psychological and physical deformity.
In terms of his appearance, Richard was born deformed. Because of the supernatural element in Elizabethan society, he was shunned by even his own mother. Historical facts also tell us of Richard III's physical deformity and how his own mother could not bear to look at him, and so he was brought up by maids. This kind of abandonment has obviously affected him negatively, and can be traced as one of the reasons he is so "determined to prove a villain". He has known no love, and therefore, can give no love to anyone in any way.
The consequences of his physical deformity, have also affected him psychologically. His mind has become twisted with evil and he uses all his strengths to achieve his evil ends.
Considering the above, it can be said that Richard is a villain because he is deformed. However, he is also deformed because he is villain. As a dramatist, Shakespeare exaggerates Richard's physical deformity to make him more villainous. In other literary works and historical accounts, Richard is also portrayed as the villainous uncle who killed the innocent princes. His psychological deformity is also increased with each murder he puts into action.
So you can really argue this both ways.