Things came to a crisis over the question of legal jurisdiction. Court and Church had different legal systems at that time. If a cleric murders someone, however, in which court should he be tried. The two courts worked in fairly similar ways, but there were important differences -- for instance, ecclesiastical courts did not have the death penalty. Henry wanted all such crimes tried in royal courts, Thomas insisted that they be tried in ecclesiastical courts. Henry was willing to accept a compromise in which the accused would first be tried in an ecclesiastical court, then, if found guilty, be defrocked and tried in the secular court; but Thomas did not regard this as as even remotely acceptable, and insisted that the accused could not be tried twice for the same crime.
Henry began actively pulling strings again, trying to get the bishops to agree to a sharply curtailed system of ecclesiastical privileges. Only Becket held out, and he was eventually forced to flee to France. Pope Alexander III began to intervene, and through his legates negotiated a compromise that allowed Becket to return in 1170.
Becket's stay on the continent seems to have done nothing to restrain his insistence on maintaining the rights and privileges of the Church; the issues changed, but his attitude did not. What follows next is somewhat obscure. The usual story is that Henry happened to say something -- the exact thing said varies from story to story -- that was taken by some of his knights as a hint to bring Thomas in. So four knights went to Canterbury and demanded that Thomas come with them to Winchester to answer for his actions. He refused, and they went back outside to get their swords, and killed him in the cathedral on December 29. We have eyewitness accounts of the murder.
Alexander III canonized him in 1173, which is very short period for canonization. But he was a martyr, which simplifies the canonization process, and as word spread, so did his veneration. He was originally buried in the crypt, but he was moved to a special shrine in 1220. That shrine stood until 1538, when Henry VIII, in the process of stealing all of the Catholic Church in England, specifically ordered the shrine destroyed (and possibly had his bones burned), and also had all mention of his name eliminated from the Cathedral and made it illegal to celebrate his feast day. No doubt he was a little uncomfortable at the thought of England's most famous pilgrimage site being devoted to a saint martyred for insisting to a King Henry that the Church had rights and privileges the king could not infringe.
Chaucer's Canterbury Tales, of course, is about a pilgrimage to Becket's shrine. Both Tennyson's Becket and T. S. Eliot's Murder in the Cathedral are about Becket's murder.
From Tennyson's Becket:
DE MORVILLE. Why, then you are a dead man; flee!
BECKET. I will not.
I am readier to be slain, than thou to slay.
Hugh, I know well thou hast but half a heart
To bathe this sacred pavement with my blood.
God pardon thee and these, but God's full curse
Shatter you all to pieces if ye harm
One of my flock!