Thomas Becket is the tragic hero in this play for two reasons. One, his friendship with Henry II, who supposedly made the comment, "Will anyone rid me of this priest?!" in the midst of his guards who took him literally. They are the ones who went to the Canterbury Cathedral and killed Becket inside.
The second reason Thomas Becket is the tragic hero in this play is the fact that he is the highest ranking church official in England, and he took his duties to the church seriously. He stood up to Henry regarding noblemen who were breaking the law of the church, and he fearlessly stood by his friends and fellow churchmen, but where Henry was concerned, Thomas did not stand up for himself. He was clever , generous, and righteous which made him popular with the common people. However, Becket believed that God's will would be done with regard to his own fate.
Remember, the word tragedy originated from the Greek word 'tragos' which means a 'goat'. do you wonder now how?
Actually, in the actual greek practice, human was made an object of sacrifice either to appease the repective god/goddess or to seek his/her benevolence. in the later time, with the progress of civilization, the greeks realized the horror and heinous crime involved with this ritual. then the human was replaced by a goat.
in this ritual, the object of sacrifice was kept at the centre while the ritual-killers surrounded it along periphery. their spears were pointed toward the centre where the object of sacrifice was already tied or affixed to the 'stake'. they would sing a ritual song and gradually as the song nears to close, they constrict the periphery and move toward the centre. thus, with the end of the song, the spears get pierced. the object, be it human or a goat, is sacrificed. As the goat had no crime, yet it is made an object of sacrifice, hence the origin of the word 'scapegoat'. and, hence too, is the reason of the origin of the word tragedy from 'tragos'.
remember about Macbeth in Shakespeare's play. when Macbeth has no way out to flee or to survive, when all his enemies have surrounded him from all sides, he utters the pang-relieving words:
"I'm tied to the stake, I cannot fly,
But bear like I must fight the course."
this imagery evokes in our mind the ancient greek practice as mentioned above.
Now, consider the case the Becket in Murder in the Cathedral. In Act II, in the murder episode in particular, he is surrounded by four murderers. he has no way out to escape. Eliot very adroitly sketches this imagery by alluding to the ancient greek practice of ritual killing. from this angle, obviously, Becket is a tragic hero, a classical tragic hero.
but there are also other points one may elaborate in details that correspond to the question whether Becket is a tragic hero. Let's have a look:
First, Becket is pre-ordained by God to be a Martyr. in this respect, he is 'a chosen seed of God'. To God, of course, he is an object of Martyr. but, as per God's design, Becket must not be aware of this. Neither Becket should want it. because christian martyrdom is only the design of God. but in his encounter with the Fourth Tempter, the reality of his heart's intention was revealed. he had thought of being a martyr perhaps in the remotest corner of his mind. once the fourth tempter is gone, becket realises his sin. if this is his hamartia then he must undergo the way of salvation that might bring him a redemption. and so does he. his redemption comes through his realization of his own remark that he made in Act II:
"Action is suffering and suffering action..."
Secondly, Becket had to die almost for no crime on his part. being loyal to the church or truthful to the religion can never be a crime. yet becket suffers in the hand of the murderers.
Thirdly, we should raise a debate with supporting arguments from various critics, whether, Becket is really a tragic hero or a religion's martyr or both? Literally, becket is a tragic hero. but eliot also projects him as a martyr. And as is the case, a religious martyr is always a sufferer, so becket must be a tragic hero too. symbolically, becket is a christ-figure, a saviour of mankind. as christ was murdered first, made a tragic hero and then a prophet, so in case of becket, he was murdered first, made a tragic hero, and then a saint.