Three of Buckingham's four grandparents were descended from Edward III of England.
Edward IV and Richard III are his first cousins once removed, so the murdered princes of the tower (and Elizabeth of York, later wife of Henry VII) were his second cousins. Henry Tudor (later Henry VII) is also his second cousin. Anne Neville, queen of Richard III, was also his second cousin.
He was executed for leading a revolt against Richard III.

Henry Stafford, 2nd Duke of Buckingham (4 September 1454 – 2 November 1483) played a major role in Richard III of England's rise and fall. He is also one of the primary suspects in the disappearance (and presumed murder) of the Princes in the Tower. Buckingham was related to the royal family of England so many different ways that he was his own cousin many times over, but his connections were all through daughters of younger sons. His chances of inheriting the throne would have seemed remote, but eventually the internecine conflicts among the descendants of Edward III of England and within the Houses of Lancaster and York brought Buckingham within striking distance of the crown. Some historians claim Buckingham's deliberate plotting to seize the throne started as early as the reign of Edward IV, and if they are correct then his elaborate and lengthy plan very nearly succeeded.

Parliament subsequently declared Edward V illegitimate offering Richard the throne, and he accepted it, becoming Richard III. After initially supporting Richard, Buckingham subsequently started working with John Morton, Bishop of Ely, in support of Buckingham's second-cousin Henry Tudor against the King, even though this placed him on the same side as his Woodville in-laws.

Buckingham's motives in these events are disputed. His antipathy to Edward IV and his children probably arose from two causes. One was his dislike for their mutual Woodville in-laws, whom Edward greatly favoured. Another was his interest in the Bohun estate. Buckingham had inherited a great deal of property from his great-great-grandmother, Eleanor de Bohun, wife of Thomas of Woodstock and daughter of the Earl of Hereford, Essex and Northampton.
Eleanor's younger sister and co-heir Mary de Bohun married Henry Bolingbroke, who eventually became Henry IV, and her share of the de Bohun estates became incorporated into the holdings of the House of Lancaster, being eventually inherited by Henry VI. When Henry VI was deposed by Edward IV, Edward appropriated that half into the Crown property under the House of York.
Buckingham claimed those lands should have been devolved to him instead, and it is likely that Richard III promised to settle the estate on Buckingham in return for his help seizing the throne. Indeed, after Richard's coronation he did award the other half of the Bohun estate to Buckingham, but it was conditional on the approval of Parliament. Historians disagree on whether this condition was in fact a way for Richard to appear to keep his promise while actually breaking it, but this may have been a motivation for Buckingham to turn against Richard.

Richard III is alleged to have consolidated his power by eliminating his brother's children, who preceded him in succession to the throne. However, there is some question about Buckingham's relationship to the disappearance of the Princes in the Tower. According to a manuscript discovered in the early 1980s in the College of Arms collection, the Princes were murdered "be [by] the vise" of the Duke of Buckingham. There is some argument over whether "vise" means "advice" or "devise," and, if the former, in what sense; for a discussion of the matter, see the article by Richard Firth Green, who discovered the manuscript, in the English Historical Review of 1981.
If Richard was responsible for killing the Princes in the Tower, the murders may have caused Buckingham to change sides. On the other hand, Buckingham himself had motivation to kill the Princes, being a Lancastrian contender for the throne with a viable claim potentially equivalent to that of Henry Tudor, depending on one's view of the legitimacy of the Tudor branch of the House of Lancaster. According to this perspective, if Buckingham killed the Princes and blamed Richard, he could foment a Lancastrian rebellion, putting the throne into play with only Henry Tudor as a rival. Indeed, a Lancastrian rebellion followed, but it was Henry Tudor who succeeded in deposing Richard III.

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Buckingham was the son of Humphrey Stafford, Earl of Stafford and Margaret Beaufort, Countess of Stafford. Four of Buckingham's first and second cousins became King of England, and two of his second cousins became Queen:

* Edward IV and his brother Richard III were Buckingham's first cousins once removed. Buckingham's father Humphrey, Earl Stafford, was son of Anne Neville (c. 1411–1480). Anne's sister Cecily, Duchess of York was the mother of Edward IV and Richard III. Edward's son Edward V was thus Buckingham's second cousin, as was the younger Edward's sister Elizabeth of York, later wife and Queen Consort of Henry VII of England.

* Henry Tudor, later King Henry VII was Buckingham's second cousin. Buckingham's mother was Margaret Beaufort (c. 1427–1474), daughter of Edmund Beaufort, 2nd Duke of Somerset. Margaret's first cousin, also named Margaret Beaufort (1443–1509) was the mother of Henry VII, the latter Margaret being the daughter of the 1st Duke of Somerset.

* Anne Neville, in line to become Queen as the wife of Lancastrian Edward of Westminster, Prince of Wales, eventually did become Queen as the wife of Richard III of England. Her paternal grandfather Richard Neville, 5th Earl of Salisbury was the brother of Buckingham's paternal grandmother (also named Anne Neville) making Buckingham the Queen's second cousin.

One can see from the ancestral chart below that two of his great-grandparents were brother and sister (John Beaufort and Joan Beaufort). This made Buckingham's parents second cousins.