Why did Richard III take the crown?

It is unikely that Edward wanted the crown as soon as he heard of Edward IV's death. If he had wanted the crown from the beginning then he'd probably have taken it far sooner than mid-June. 

Instead events unfolded slowly:

1) The council appointed Richard as Protector and this was supported by Edward IV's household men (led by Hastings)

2) The administration of government continued normally with few changes in personnel

3) There was six whole weeks between the arrest of Rivers on 30th April and the execution of Hastings on 13th June - six weeks with no sign of Edward wanting the crown. This strongly suggests that Richard was not ambitious for power the moment Edward IV died.

Richard said twice that his actions were driven by Woodville threats to his life and power. He justified the arrest of Rivers and Grey in late April by claiming they were plotting to stop him taking a leading part in the young King's council. 

Did the Woodvilles want to stop Richard playing a leading role in the council? Almost certainly, yes. The Crowland Chroncile implies the Woodvilles wanted to dominate Edward V's government instead of there being a broad council including Richard. So the Woodvilles threatened Richard's place in power. There was no previous hostility between them though. Rivers was taken by surprise when arrested by a man he thought his friend. However, Richard was heavily influenced by Buckingham and Hastings  who had rivalries with members of the Woodvilles

Did the Woodvilles plan to murder and utterly destroy Richard?: If they did, we must look at Richard's actions: arresting Rivers, taking the Woodvilles lands and positions and giving their wealth to Buckingham. These actions created an enmity between Richard and the Woodvilles, symbolised by Elizabeth Woodville and most of her family taking sanctuary at Westminster Abbey. By early June Richard must have realised what this enmity would mean in the future. When Edward V could rule himself he would recall his relatives to positions of power and they would take revenge on Richard. This is probably why he striked for the crown in June. 

This may have been an important motive. Richard's possession of his vast northern territory wasn't permanently secure. They had been granted by his brother and could be taken away. In particular his hold on parts of the former Neville family land was vulnerable. The real heir to these lands was George, Duke of Bedford, but Bedford couldn't inherit because his father had been attainted for treason. But Bedford died without children on 4th May 1483, ending the attainder and changing Richard's hold on the land. He would keep it until he died but then it would go to the next Neville heir, The Latimers, not to Richard's son. 

So was Richard motivated to take the crown to safeguard his lands? 

It's a possible motive. It would be even more probable if we knew when Richard found out about Bedford's death. If he didn't find out until late May or early June, this would make sense of the sudden rush of events and his ruthless execution of Hastings. 

Richard said he was the rightful king because everyone more closely related to Edward Iv was barred from the throne, either by illegtimacy (Edward's children) or by Act of Attainder for treason (Clarence's son). 

Did Richard believe that his nephews were illegtimate, and so believe that he was rightfully king?

OR

Did Richard fake the story as a legal excuse for taking the crown? 

There are strong arguments for it being fake. Its appearance out of the blue at an extraordinarily convenient moment raises suspicion. More importantly, bastardy need not have stopped Edward V become king because the coronation would have wiped out illegtimacy. Richard ignored this possibility, which suggests his priority was his own power and position. FInally, people's reactions suggest widespread doubt about the story. Many gentry rebelled against Richard because they did not believe in his right to be king.  

He may have been motivated by a sense of duty. In the 1450s his father Duke of York had been convinced England needed him to end the political crisis. Perhaps Richard saw himself as the man for a crisis, guiding England through the potential problems of Edward V's minority. His claim to the throne emphasised his devotion to the common good and to righting past failures. Such a sense of duty would fit with his deep religious faith and serious approach to government when he was king. It may help explain his determination to be Protector and perhaps his decision to take the crown. 

However, he must have known that deposing Edward V would lead to instability and possibly rebellion. 

It seems that Richard decided to take the crown by the 10th of June, when he wrote to the city of York for armed support. 

Did he make this decision simply out of desire to be king?

Richard has often been portrayed as motivated by power because of the ruthlessness with which he seized the crown. William Lord Hastings would not countenance Richard becoming king so on 13th June Richard had Hastings arrested and executed without trial. A fortnight later Rivers and Grey were executed without trial. The shock of this violence silenced potential opposition.

Conclusion:

It certainly looked as though he were solely interested in power, but behind these actions probably lay either fear of Woodville revenge and the loss of his nothern lands or the belief that he was the rightful king and a sense of duty.