Richard, Duke of Gloucester, is bored.
By the time Richard was eighteen or nineteen years old, he was leading troops into battle and gaining victories for his brother, Edward, King of England, during the "War of the Roses"—the war between the Lancastrians and the Yorkists, two sides of Richard's family, the Plantagenets. For the past ten or twelve years of his life, Richard has only known war. He was good at it, and he enjoyed it.
At the beginning of Shakespeare's Richard III, King Edward is safely on the throne, England is at peace, and Richard has nothing to do. In his opening soliloquy, Richard tells us everything we need to know about his motivations for wanting to be King.
RICHARD. Now is the winter of our discontentMade glorious summer by this son of York;And all the clouds that lour'd upon our houseIn the deep bosom of the ocean buried (1.1.1–4).
The War of the Roses is over.
RICHARD. ...And now, instead of mounting barded steedsTo fright the souls of fearful adversaries,He capers nimbly in a lady's chamberTo the lascivious pleasing of a lute (1.1.9–13).
This might be a good thing for everybody else at court, but while they're all dancing and singing around the castle, Richard is growing increasingly unhappy and discontented.
RICHARD. ...But I, that am not shaped for sportive tricks,Nor made to court an amorous looking-glass;I, that am rudely stamp'd, and want love's majestyTo strut before a wanton ambling nymph...
Why, I, in this weak piping time of peace,Have no delight to pass away the time,Unless to spy my shadow in the sunAnd descant on mine own deformity...(1.1.14–17, 24–27)
Richard mocks his own deformity, remarks on his inability to be loved, and tells us that he prefers his own company rather than engage in activities that he considers beneath his dignity.
Richard might simply be rationalizing that he doesn't want what he can't have anyway, but he seems to be honestly repulsed by the delights that he sees going on around him.
Shakespeare's character invariably tell the truth in their soliloquies, so we should take Richard at his word.
RICHARD. ...And therefore, since I cannot prove a lover,To entertain these fair well-spoken days,I am determined to prove a villainAnd hate the idle pleasures of these days (1.1.28–31).
Richard says that he's going to be a villain. There's no reason for us to doubt him. He even tells us that he's already begun behaving villainously against his own brother, Clarence.
Richard says nothing about ambition in this soliloquy. The word "ambition" appears only once in the play, in act 3, scene 7, when Richard denounces ambition while he's trying to impress the Lord Mayor and some citizens with his false piety and religious zeal.
Later in the scene, Richard muses about what will happen when both of his brothers, Clarence and Edward, are dead.
RICHARD. ...And, if I fail not in...
my deep intent,Clarence hath not another day to live;Which done, God take King Edward to His mercy,And leave the world for me to bustle in. (1.1.153–156)
Richard says nothing about wanting to be king. For now, Richard seems content simply to set events in motion, if for no other reason than to see how much he can manipulate the people around him and see how far he can get.
Richard doesn't talk about being king until act 3, scene 1.
RICHARD. (to Buckingham) ...And look when I am king, claim thou of meThe earldom of Hereford, and all the movablesWhereof the King my brother was possessed (3.1.197–199).
It appears that Richard doesn't need a driving, motivating force like ambition because Richard believes that it's simply inevitable that he'll be king. He'll just keep putting his villainous plans into effect until he becomes king.
Richard is playing a game that he intends to win. And if he loses?
RICHARD. ...I have set my life upon a cast,And I will stand the hazard of the die (5.5.9–10).