The Duke of Buckingham is Richard's wingman for most of the play. He's greedy and he's willing to lie, cheat, and steal to help his pal Richard get the crown.
Like a lot of other characters in the play, Buckingham is duped into thinking Richard will reward his loyalty. When Richard promises to give him the earldom of Hereford once he becomes king, Buckingham believes him (3.1.16). Big mistake.
Things are swell for Buckingham until he waffles when Richard orders him to murder the young princes in Act 4, Scene 2. (Richard pointedly says "I wish the bastards dead" and Buckingham says something like "Gee, can I get back to you on that?") Buckingham's hesitation is an important moment in the play, because it signals that Richard has crossed a big line by plotting the deaths of innocent children. (If it makes Buckingham uneasy, you know it's downright wicked, because Buckingham isn't exactly a nice guy.)
Since he knows he's ticked off Richard, Buckingham hightails it to Wales (4.2.19), where he hides out until he can join forces with Edmond against Richard (4.3). But just like all the other murderous, scheming characters in the play, Buckingham gets his comeuppance. Catesby reports that he's captured and sent to the chopping block (4.4.5). That's not the last we see of Buckingham, though. He joins the parade of Ghosts that return to haunt Richard's dream in Act 5, Scene 3.