This is what I understand to be the crime for which they wanted to kill Him.
57 And those who had laid hold of Jesus led
Him away to Caiaphas the high priest, where the scribes and the elders were assembled. 58 But Peter followed Him at a distance to the high priest’s courtyard. And he went in and sat with the servants to see the end. 59
Now the chief priests, the elders, and all the council sought false testimony against Jesus to put Him to death, 60 but found none. Even though many false witnesses came forward, they found none. But at last two false witnesses came forward 61 and said, “This
fellow said, ‘I am able to destroy the temple of God and to build it in three days.’”62 And the high priest arose and said to Him, “Do You answer nothing? What
is it these men testify against You?” 63 But Jesus kept silent.
And the high priest answered and said to Him, “I put You under oath by the living God: Tell us if You are the Christ, the Son of God!” 64
Jesus said to him, “It is as you said. Nevertheless, I say to you, hereafter you will see the Son of Man sitting at the right hand of the Power, and coming on the clouds of heaven.” 65 T
hen the high priest tore his clothes, saying, “He has spoken blasphemy! What further need do we have of witnesses? Look, now you have heard His blasphemy! 66 What do you think?” They answered and said, “He is deserving of death.” Matthew 26:57-66 NKJV
Here, in Luke's gospel, chapter 23:1-4 NKJV, we can read this. Notice that Pilot doesn't find that a crime was committed even after Jesus admitted to being the King of the Jews.
1 Then the whole multitude of them arose and led Him to Pilate. 2 And they began to accuse Him, saying, “
We found this fellow perverting the nation, and forbidding to pay taxes to Caesar, saying that He Himself is Christ, a King.” 3 Then
Pilate asked Him, saying, “Are You the King of the Jews?” He answered him and said, “It is as you say.” 4 So
Pilate said to the chief priests and the crowd, “I find no fault in this Man.”
Same thing here in John's gospel, Chapter 18:28-38 NKJV.
28 Then they led Jesus from Caiaphas to the Praetorium, and it was early morning. But they themselves did not go into the Praetorium, lest they should be defiled, but that they might eat the Passover. 29 Pilate then went out to them and said, “What accusation do you bring against this Man?” 30 They answered and said to him, “If He were not an evildoer, we would not have delivered Him up to you.” 31 Then Pilate said to them, “You take Him and judge Him according to your law.” Therefore the Jews said to him, “It is not lawful for us to put anyone to death,” 32 that the saying of Jesus might be fulfilled which He spoke, signifying by what death He would die. 33 Then
Pilate entered the Praetorium again, called Jesus, and said to Him, “Are You the King of the Jews?” 34 Jesus answered him, “Are you speaking for yourself about this, or did others tell you this concerning Me?” 35 Pilate answered, “Am I a Jew? Your own nation and the chief priests have delivered You to me. What have You done?” 36 Jesus answered, “My kingdom is not of this world. If My kingdom were of this world, My servants would fight, so that I should not be delivered to the Jews; but now My kingdom is not from here.” 37
Pilate therefore said to Him, “Are You a king then?” Jesus answered, “You say rightly that I am a king. For this cause I was born, and for this cause I have come into the world, that I should bear witness to the truth. Everyone who is of the truth hears My voice.” 38 Pilate said to Him, “What is truth?” And when he had said this, he went out again to the Jews, and said to them,
“I find no fault in Him at all.
Then here in chapter 19:1-16 NKJV they finally convince Pilate.
1 So then Pilate took Jesus and scourged
Him. 2 And the soldiers twisted a crown of thorns and put
it on His head, and they put on Him a purple robe. 3 Then they said,[
a] “Hail, King of the Jews!” And they struck Him with their hands. 4 Pilate then went out again, and said to them,
“Behold, I am bringing Him out to you, that you may know that I find no fault in Him.” 5 Then Jesus came out, wearing the crown of thorns and the purple robe. And
Pilate said to them, “Behold the Man!” 6 Therefore, when the chief priests and officers saw Him, they cried out, saying, “Crucify
Him, crucify
Him!”
Pilate said to them, “You take Him and crucify Him, for I find no fault in Him.” 7 The Jews answered him, “We have a law, and
according to our law He ought to die, because He made Himself the Son of God.” 8 Therefore, when Pilate heard that saying, he was the more afraid, 9 and went again into the Praetorium, and said to Jesus, “Where are You from?” But Jesus gave him no answer. 10 Then Pilate said to Him, “Are You not speaking to me? Do You not know that I have power to crucify You, and power to release You?” 11 Jesus answered, “You could have no power at all against Me unless it had been given you from above. Therefore the one who delivered Me to you has the greater sin.” 12
From then on Pilate sought to release Him, but the Jews cried out, saying, “If you let this Man go, you are not Caesar’s friend. Whoever makes himself a king speaks against Caesar.” 13 When Pilate therefore heard that saying, he brought Jesus out and sat down in the judgment seat in a place that is called
The Pavement, but in Hebrew, Gabbatha. 14 Now it was the Preparation Day of the Passover, and about the sixth hour. And he said to the Jews, “Behold your King!” 15 But they cried out, “Away with
Him, away with
Him! Crucify Him!”
Pilate said to them, “Shall I crucify your King?” The chief priests answered, “We have no king but Caesar!” 16 Then he delivered Him to them to be crucified. So they took Jesus and led
Him away.