The historical events behind Eliot's verse drama pertain to an on-going quarrel between King Henry II--who is reported to have shouted, "Will no one rid me of this turbulent priest?"--and Thomas Beckett, who changed his extravagant life for one of ecclesiastical devotion once he was made Archbishop of Canterbury. The quarrel centered around a disagreement over what law had authority over clergy who committed secular (non-religious) crimes.
King Henry said that even clergy were subject to the law of the land of England, especially since longstanding custom in England supported the practice of secular law and courts ruling over clergy's secular crimes. This sounds reasonable to a modern reader, but in the 1100s, religion and the Church had great and powerful authority--and seemingly desired more.
Archbishop of Canterbury Thomas Beckett was newly appointed by King Henry II to be his representative and advocate in the Church of England, a decision that garnered much disfavor because Becket, a military leader, had never been in Holy Orders. Becket's position was that clergymen, even those who had taken lower orders of the clerical hierarchy, were exempt from secular law and could only be tried and punished within ecclesiastical, or Church, courts.
This disagreement, a serious one since approximately one-fifth of the male population of England was in clerical orders (and occasionally given to secular crime ...), was escalated because the crime rate had increased and criminals were going unpunished because, when clergymen, they claimed the right to be tried in Church courts.
This was compounded by Becket's determination to reclaim previously confiscated Church lands and by the complaints this land reclamation effort engendered from the present occupants of the land. Becket and his family were finally exiled after long and complicated disputes over the authority to rule in criminal offenses, a dispute that finally fell to Henry's victory, and the restoration of the customs from his grandfather's day and the recognized authority of the Crown's courts over all secular crime, even that committed by clergymen.