Edward, far from the frail old man the play depicts, died unexpectedly in his early forties. Richard was not a hunchback, and wasn't even at court when Edward died.
Turns out that Richard did have scoliosis. So the hunchback theory is not that far off.
Richard also married Anne much earlier than the play suggests — they had a ten-year-old son by the time Richard was crowned.
Recent records found in Portugal also reveal that Richard was trying to arrange marriages to Portuguese royals for both himself and teenaged Elizabeth, contrary to claims that he planned to marry her himself. (Of course, this was done after rumors that Richard had planned to marry her had already caused a great deal of negative outcry, leading the King formally to deny any such intention.)
The idea was to pair himself off with the King of Portugal's sister (who unfortunately had a sincere religious vocation and would have none of it) and Elizabeth with the King's cousin, Manuel, Duke of Beja. If the latter plan had gone through, she would still have wound up a Queen - Manuel inherited the throne when the King's heir died without issue.
Also, Richard never himself formally accused his mother of being an adulteress; his claim to the throne was based on his brother's marriage to Elizabeth Woodville being illegal due to a previous (secret) marriage to Eleanor Butler. Whether or not that was proven, Edward's long pattern of horn-dog behavior made it easy for people to believe. On the other hand, Richard's adherents (notably Dr. Ralph Shaw (or Shaa), in a public sermon given at St. Paul's Cross) had already floated rumors of the Duchess of York's misconduct, hinting that Richard resembled his father far more than Edward did.
Also, the fate of the Princes was never certainly determined, and some years after Richard's death, Perkin Warbeck came forward claiming to be the younger prince (a claim supported by the princes' aunt Margaret of Burgundy). What is proven is that all the other heirs of the House of York were exterminated by Henry VII, except for the few killed by his son Henry VIII. The only one they didn't get was Margaret, who was safely out of their reach in Burgundy.
Buckingham's rebellion was intended to put Buckingham himself on the throne, not restore the throne to Edward V. It is possible, of course, that he knew that the young King was already dead; alternatively, the idea of a Yorkist and Woodville king might have been utterly distateful to him, a long-time adherent of the Lancastrian cause and an embittered oppoent of the Woodville faction. Buckingham quickly transferred his support to Henry Tudor, however, perhaps having been convinced of the impossibility of receiving the support either of the Yorkist nobles (as he was rebelling against the Yorkist King Richard) or of the Lancastrians (as he had been for so long identified with Richard's interests).
Also left out were the facts that Elizabeth and her family, the Woodvilles, were not of noble lineage and were very unpopular with both nobles and commoners. The elder prince was educated by her brother and many feared that as king, Edward V would be a Woodville pawn. Also, that both of England's previous civil wars had begun because of a child-king holding the throne, and the prospect of another one terrified the beleaguered country. Richard, as a grown man with a proven record of good governing, an heir already in place, and a reputation for being no friend of the much-disliked Woodvilles, looked like a great solution to all those problems, which is why Parliament offered him the throne. (Well, that and the Yorkist army investing London.)
As for the body count, Richard's was fairly low compared to many monarchs of the time period.
And "poor, perjur'd Clarence" had been part of an armed rebellion against his brother Edward, among other increasingly lunatic stunts until his murder of a servant girl drove the elder Edward to order his death. Richard argued against Clarence's execution despite his previous feuds with Clarence, and when the verdict was announced Richard left the court for his estate in Middleham. Contrary to the claims of the play, there was no belated commuting of the sentence — Edward wanted Clarence dead, for reasons that had nothing to do with prophecies.
Product Placement: In the 1995 film, Ian McKellen's Richard is often seen smoking "Abdulla" non-nicotine herbal cigarettes, which hadn't been made for decades. Yes, they are the real deal, and getting the last five or six packs in existence was a Crowning Moment of Awesome for the props department.