Kings are generally thought to be beautiful, God-like humans from a certain genealogy. They are placed at the throne due to whom they were born to and/or who they marry. Richard of Glouster is born of the royal family but, he does not possess the qualities of a King to be legitimate according to Shakespeare's Richard III.
Richard and his brothers are all entitled to the throne if their father dies. By killing his two brothers, Richard is the only person in his immediate family entitled to the throne. Why must he kill his brothers so that he may gain the crown? Shakespeare depicts Edward and Clarence as far more worthy of the crown. They possess traits of a King which Richard does not possess.
Richard states in his opening speech "Unless to spy my shadow in the sun/And descant on mine own deformity./ And therefore since I cannot prove a lover" (1.1.26-28).
In this speech Richard admits he cannot be a lover because he is ugly with deformity. This imperfectness does not fit the perfect qualities known to Kings. His brothers, on the other hand, do possess the ability to love and both were married or betrothed.
Edward even had two women to love! Some may say that Edward was illegitimate because of his other lover, but, is it better and more respectable to not be able to have love or to have several loves? A King should be able to love to rule a Kingdom otherwise he may be a ruthless killer as Shakespeare demonstrates in the play.
Gaining the hand of a woman not only makes a King look worthy to love but also is used for political gains. Richard himself demonstrates this as he ruthlessly goes after Ann after killing her lover and her Father.