The story, as related by Sir Thomas More and then William Shakespeare, involved two attempts to kill Edward and Richard, with the second being successful.
On July 17th 1483, Richard appointed a long-time ally and supporter named Robert Brackenbury to the position of Constable of the Tower. This was now the man charged with the day to day care of the two royal boys.
Three days later, Richard III left Windsor to begin the tour of his realm. On July 29th, the king reached Gloucester. Now safely distanced, he dispatched a rider named John Green with a verbal message for Brackenbury back at the Tower of London.
The message asked the Constable to murder the children. But Brackenbury knelt before 'Our Lady of the Tower' (the Virgin Mary in the chapel) and swore that he would not do it, even if it meant his own death for refusing.
John Green rode away with this message. He caught up with his king at Coventry, on August 15th, and told him about Brackenbury's refusal. Richard apparently then went to the toilet. He sat there, muttering to himself, while he engaged in a long number two. His monologue was along the lines of 'who will rid me of these turbulent princes?'
A young boy - a page of the Chamber - overheard him and suggested that James Tyrrell would do the job. Tyrrell was the man who slept each night across the threshold to Richard's bedroom door. He wasn't actually there as a draught excluder. He was to ensure that no-one interrupted the king's sleep. Tyrrell was naturally looking for promotion.
Bowels cleared, Richard thanked the page for his recommendation of a murderer and promptly sent for James Tyrrell. This was a man with a criminal record, who jumped at the chance to do this thing. He was sent to London, ostensibly to collect some robes, so that Richard's son could be proclaimed Prince of Wales (in Yorkshire) in style.
(Slight verifiable discrepancy here. Tyrrell did indeed go to London for the robes, but not on August 15th from Coventry. The King's Wardrobe Accounts state that he left on August 30th from York.)
Sometime in early September (one historian, Alison Weir, worked out that it was September 3rd 1483), Tyrrell - with his accomplices John Deighton and Miles Forrest - arrived at the Tower of London. Brackenbury, after swearing that he'd rather die than see harm come to those children, and after undertaking a job which restricted all access to the boys, apparently just opened the door for the asking of three assassins.
Tyrrell and Deighton promptly smothered Edward V and Richard of Shrewsbury with pillows as they slept.
The bodies were then carried down the spiral steps of the north-eastern tower of the White Tower, across the open courtyard, down the slope and into the gateway of the Bloody Tower. There an open grave had been prepared at the foot of the main staircase. The boys were buried with their golden robes (which must also have been conveyed to the spot), only to be uncovered in 1674, when restoration work was being undertaken in that area.
All this without the three hundred denizens of the Tower of London seeing them or suspecting a thing.