Richard III might or might not have had the Princes in the Tower killed (the evidence is far from conclusive, and there is at least one other suspect with equal opportunity and arguably a greater motive). He certainly did not commit most of the other crimes with which Shakespeare charges him. On the other hand, he invented the custom of legal aid for the indigent, reformed the practice of bail, instituted the system of royal messengers which became the British postal system, and abolished press censorship- and all in a reign of approximately two years.
Many kings have reigned far longer with many fewer accomplishments. Even at the worst, Richard was less bloody a monarch and in fact a far more benign one than most (contemporary historian John Rous praised him as a "good lord" who brought "oppressors of the commons" to heel and had a "great heart").
The need of Henry Tudor- an adventurer with no real claim to the throne- to discredit him resulted in a deluge of Tudor propaganda (including Shakespeare's play) which gave a good king one of the worst names in British history. Perhaps now that historic wrong will be righted.
ADDENDUM: The Chapter atYork Minster has released a statement renouncing all claim to King Richard's remains, and commending them to the good folk of Leicester.
It's a shame that Richard's expressed wish concerning his burial place will be disregarded, but that would seem to be that.