How did Richard usurp the throne?

A cancelled Coronation

With the story of the Princes' illegitimacy circulating, Richard, Duke of Gloucester cancelled Edward V's coronation and later forced Elizabeth Woodville to hand over her other son, Richard, Duke of York.

to lodge with his brother in the Tower of London. 

Beheading of Hastings

Richard then called a meeting of loyal Councillors. Among them was Lord Hastings who was a loyal supporter of Edward IV and was likely to resist Richards attempts to seize the throne. Mid meeting, on Richards orders, Lord Hastings was arrested in the council chamber and beheaded on the spot.

The son of George, Duke of Clarence

Even if Edward IV's sons had been illegitimate, their cousin Edward, Earl of Warwick was now the real heir of Edward V. (George, Duke of Clarence was brother to Edward IV and father of Edward, Earl of Warwick). However, Richard was one step ahead and already had the Earl of Warwick in custody so he couldn't challenge Richards own claim to the throne.

A powerful and cunning man

Richard had proven himself to be a powerful and cunning man. On the 22nd June, 1483 a proclamation was made that Richard was the rightful King, that Edward IV's marriage was illegitimate and that the Princes were bastards. 

This was ratified by parliament, Richard had himself crowned King on the 6th off July 1483 with parliaments blessing. His title, king of England, was confirmed by parliament in their session of 1484.

 

Summary of how Richard usurped the throne in 24 words:

Richard rids of all the kids and locks them in the tower,

with Hastings arrested, parliament blessed him,

and now he is in power.