Why did King Henry II want the Archbishop of Canterbury, Thomas Becket, killed?
Actually he only wished him dead and willing servants made it happen. They had long and complex disagreement about aspects of secular governance.
Henry II wanted the same arrangement that his grandfather, Henry I, had enjoyed where he was able to appoint Bishops and a number of other arrangements that had become the custom in England. There was a long negotiation process that eventually Henry's wishes came to pass but it was a long and frustrating time.
When Beckett was killed The King had to do penance for wishing Beckett's death, but he still ruled. The Penance involved a large number of Church official striking him with a set number of blows.
After Henry II, the secular justice system was more codified. A number of things were written in the "Constitutions of Clarendon". Which the Bishops and Beckett agreed to.
The Pope politically needed Henry in the Pope's disputes with the Holy Roman Empire. Consequently the Pope was open to compromise and willing to let the disagreement be resolved over time and to Henry's favor.