It is not known for certain who killed Edward V and Richard, duke of York, but it was most likely one of Richard III's supporters who did so. It is unlikely that Richard would have done so himself, it would be better to be distant from the crime. James Tyrell, one of Richard's trusted lieutenants, was considered in the early sixteenth century to have been the likely culprit. He supposedly confessed to killing the princes, but no original of this confession survives.

Ricardians have often claimed that Henry VII or his allies killed the princes, but this seems unlikely. There was no reason for Richard to leave them alive and almost every reason to kill them. They were a threat to his claim to the throne and were a focus for rebellion. Less than a month after he became king, there was a plot to free the princes. In autumn 1483 the duke of Buckingham, who had assisted Richard's coup earlier that year, turned on the king and tried to overthrow him. Buckingham's motives have been debated, but it is likely that he aimed to restore the princes. He only had a weak claim to the throne himself and had been awarded an array of lands and offices by Richard, so self-interest or dissatisfaction was an unlikely cause. It's even less likely he backed Henry VII's claim, as this was really quite weak, though the two did know each other as children. If the princes were not already dead before Buckingham's rebellion, Richard would have killed them soon after, the uprising proving that it was too dangerous to keep them alive.

The reason for killing the princes is much more certain. As the sons of Edward IV, they had a much stronger claim to the throne than Richard did as Edward's brother. Richard tried to weaken their claim by alleging that they were illegitimate, but his claims don't seem to have been widely accepted. The existence of rival claimants to the throne was a major threat to his rule.

The damage that could be done by rival claimants can be seen in the trouble that Lambert Simnel and Perkin Warbeck caused for Henry VII. Simnel claimed to be the earl of Warwick, nephew of Edward IV. This was a very ill-judged lie as the real Warwick was still alive and held prisoner by Henry. When Simnel's claims became known, Henry seem paraded Warwick in public to prove that he was still in his custody and that Simnel was a liar. However, Simnel still attracted some support, including a few old Ricardians and another nephew of Edward IV, John de la Pole, the earl of Lincoln. Simnel and his supporters led an army into England and fought Henry VII at the battle of Stoke in 1487.

Warbeck claimed to be Richard, Edward's younger son. Warbeck received wide support in Europe, being received as the real Richard in the Holy Roman Empire, Burgundy, and Scotland. He led attacks on Ireland, followed by a raid with Scottish support into northern England, and finally he besieged Exeter before his support collapsed and he fled. He was eventually imprisoned and executed following an escape attempt.

Sources:

Rosemary Horrox, Richard III: a Study of Service (Cambridge, 1989).

Michael Hicks, The Wars of the Roses (London, 2010).