At that time, the church had made itself immune from secular law and was getting away with all sorts of wrong doing, including murder. Henry believed that the same law and justice should apply to everyone, church and state.

By appointing his friend and chancellor Thomas as archbishop, Henry hoped that these infringements and abuses of the law would stop. He was wrong.

As soon as Thomas was made archbishop, he resigned as chancellor to Henry and devoted himself to the interests of the church - in particular to Canterbury. Thomas would no longer help the king. He opposed Henry about matters of law at every turn.

Henry found allies to help him. These allies understood both civil and canon law. They helped Henry propose a law so that when a churchman had been found guilty of an offence and ‘degraded’ by the church, the churchman should then be handed over for punishment in a secular court and no longer be under the protection of the church.

As their relationshop deteriorated, so Henry resolved to ruin his former friend. He levelled accusations at Thomas about his former chancellorship. He tried to get him convicted, but Thomas fled to the Continent from Northampton before sentence could be carried out. He took refuge in the Cistercian abbey of Pontigny. He spent his time studying and writing many letters to his friends and enemies.

Thomas became increasing bitter in exile and excommunicated his enemies. The king himself was threatened with the same treatment. The pope annulled the excommunications. The dispute between king and archbishop continued for six years. Thomas was trying the patience of his friends as well as his enemies.

Now Henry wanted to secure the succession of his son - also named Henry - to the throne. He wanted his son to be crowned king while he, his father, was still alive. This had never happened in England before (Stephen had tried and failed to get his son Eustace crowned in 1152). Henry had a problem. The right of performing the coronation was the prerogative of the archbishop of Canterbury - Thomas Becket. And he was in exile and the sworn enemy of the king. So Henry asked the pope to grant permission for the Archbishop of York to perform the coronation instead. While the pope hesitated, Henry went ahead without authority and his son was crowned on 14 June 1170 by Archbishop Roger of York.

Becket was outraged. And yet, about a month after the coronation, Henry and Thomas met at La Ferte Barnard (north central France) and went through a form of conciliation. Why these two obstinate men suddenly came together in friendship is not easy to understand. But Becket’s actions suggest that he could take more effective measures against the bishops who had taken part in the coronation. With the authority of the pope, he served sentences of suspension on the offending bishops, and added excommunication on those who had especially injured him.

He landed at Sandwich on December 1. Biographers emphasize that he was warmly received by the clergy and people of Canterbury. Those in authority, however gave him anything but a warm welcome. On Christmas day, Thomas publicly denounced and excommunicated his enemies from the pulpit of Canterbury cathedral.

Three of the suspended bishops went to the king in Normandy and complained (and probably exaggerated) about Becket’s relentless conduct. Their report threw the king into a paroxysm of rage and drew from him the rash words which led to Becket’s martyrdom.

Without waiting for the king’s temper to cool, four of his knights hastened to England, and, before the king could get a message to his knights to prevent violence, Thomas Becket was murdered in his cathedral.

Thomas Becket was venerated as a martyr throughout Christendom. Within two years of his death, he was canonized by Pope Alexander III on 21 February 1173.

And that is how Canterbury became a place of pilgrimage.
References :
Domesday Book to Magna Carta 1087 - 1216, Poole, The Oxford History of England.