At Jesus’ trial, He was questioned by Caiaphas the high priest:
Matthew 26:63-66 (RSV) And the high priest said to him, “I adjure you by the living God, tell us if you are the Christ, the Son of God.”  Jesus said to him, “You have said so. But I tell you, hereafter you will see the Son of man seated at the right hand of Power, and coming on the clouds of heaven.”  Then the high priest tore his robes, and said, “He has uttered blasphemy. Why do we still need witnesses? You have now heard his blasphemy.  What is your judgment?” They answered, “He deserves death.”
The rationale for blasphemy couldn’t sensibly be (from the 1st century Jewish pharisaicl perspective) that Jesus was merely claiming to be the Messiah (citing Daniel 7:13-14, a well-known messianic passage).
That alone wouldn’t be blasphemy (though it could very well be a serious falsehood, if the claimant was in fact not the Messiah). So why did Caiaphas charge Him with blasphemy? Perhaps it’s because Jesus said “you will see the Son of man seated at the right hand of Power.” See:
Daniel 7:13 I saw in the night visions, and behold, with the clouds of heaven there came one like a son of man, and he came to the Ancient of Days and was presented before him.
Psalm 110:1 The LORD says to my lord: “Sit at my right hand, till I make your enemies your footstool.”
But what doesn’t make sense is that both passages present the Messiah in [personal] contrast to God [the Father] (as the second person of the Trinity, so we claim). There is such a person who sits next to God, in the Hebrew Scriptures: the Messiah. Jesus claimed to be the Messiah (Christ) in this same passage. I would love to ask Caiaphas, then, if I could: “why do you assume that Jesus was not the Messiah? Why could He not possibly be the Messiah? How do you know that He was not? And why did you accuse Him of blasphemy?”
Hence, Expositor’s Greek Testament writes, in harmony with my reasoning:
Was it blasphemy for a man to call Himself Messiah in a country where a messiah was expected? Obviously not. It might be to call oneself Messiah falsely. But that was a point for careful and deliberate examination, not to be taken for granted. The judgment of the high priest and the obsequious vote of the Sanhedrim were manifestly premature.
Likewise, Barnes’ Notes on the Bible:
If he had not been the Messiah, the charge would have been true; but the question was whether he had not given evidence that he was the Messiah, and that therefore his claims were just. This point – the only proper point of inquiry – they never examined. They assumed that he was an impostor, and that point being assumed, everything like a pretension to being the Messiah was, in their view, proof that he deserved to die.
I think the odd and unjust behavior is explained by the fact that the council was already prejudiced against Him, based on false testimony, and wanted to kill Him, and would accept any pretext in order to do so. They weren’t thinking rationally or fairly. Hence it says in context:
Matthew 26:59-60 Now the chief priests and the whole council sought false testimony against Jesus that they might put him to death,  but they found none, though many false witnesses came forward. . . .
See similar passages:
Matthew 27:1 When morning came, all the chief priests and the elders of the people took counsel against Jesus to put him to death; (cf. Mk 14:55)
Luke 22:2 And the chief priests and the scribes were seeking how to put him to death; for they feared the people.
John 11:45-53 Many of the Jews therefore, who had come with Mary and had seen what he did, believed in him;  but some of them went to the Pharisees and told them what Jesus had done.  So the chief priests and the Pharisees gathered the council, and said, “What are we to do? For this man performs many signs.  If we let him go on thus, every one will believe in him, and the Romans will come and destroy both our holy place and our nation.”  But one of them, Ca’iaphas, who was high priest that year, said to them, “You know nothing at all;  you do not understand that it is expedient for you that one man should die for the people, and that the whole nation should not perish.”  He did not say this of his own accord, but being high priest that year he prophesied that Jesus should die for the nation,  and not for the nation only, but to gather into one the children of God who are scattered abroad.  So from that day on they took counsel how to put him to death.
So when He claimed to be the Messiah and applied Daniel 7 and Psalm 110 to Himself, that was enough for Caiaphas to conclude that He was a blasphemer: either by falsely claiming to be the Messiah or God or both, or improperly expressing exclusively divine prerogatives. But in pharisaical mainstream Judaism the Messiah was not God. So it’s odd that Caiphas said to Jesus: “tell us if you are the Christ, the Son of God.” In any event, it seems to me that He is indeed claiming to be God, and the Jews react precisely as they did in John 5 and 10:
John 5:17-18 But Jesus answered them, “My Father is working still, and I am working.”  This was why the Jews sought all the more to kill him, because he not only broke the sabbath but also called God his Father, making himself equal with God.…
John 10:30, 33 “I and the Father are one.”…  The Jews answered him, “It is not for a good work that we stone you but for blasphemy; because you, being a man, make yourself God.”
The Jews of that time certainly seemed to think that “Son of God” and calling God “Father” / Abba was a blasphemous declaration of Godhood. But they didn’t regard the Messiah as God incarnate.
The whole thing doesn’t add up, any way you look at it. Caiaphas and the council simply wanted to put Jesus to death, and used any pretense or false pretext in order to do it.
Photo credit: Christ before Caiaphas (early 1630s), by Matthias Stom (fl. 1615-1649) [public domain / Wikimedia Commons]