So how exactly did Judas betray Jesus??
He didn't. Judas did exactly as planned by God. Even Jesus knew this when He told Judas to go and do what he had to do quickly, when He was in the upper room after sharing the last supper with His Disciples. That included Judas.
Jesus would have not been nailed to the cross were it not for Judas. Jesus was God in flesh. He chose Judas knowing who he was from the beginning as one who would help Jesus fulfill His mission on earth; to die and take the sins of the world upon Himself on the cross.
The Book of Luke chapter 22 This is why Judas shared in the last supper prior to leaving, as Jesus instructed him to do min you, and telling the temple guard where to find Him later in the garden of Gethsemane.
Judas actually repented of his sin, see the
Book of Matthew chapter 27. And then he hanged himself because he could not bear to live with what he'd done.
However, remember on the cross? Jesus forgave everyone responsible for His being there? That would include Judas. "Father forgive them for they know not what they do."
That would include Judas because he did not know until after at Jesus arrest that he had betrayed innocent blood. Remember, the elders of the temple wanted Jesus arrested. And sought to have him executed by Rome, because it was too close to Passover for them to stone Jesus to death under the charge of blasphemy, so they sought Rome's assistance to execute Jesus under the charge of sedition.
See the
Book of Luke chapter 23: Then the whole company of them arose and brought him before Pilate. 2 And they began to accuse him, saying, “We found this man misleading our nation and forbidding us to give tribute to Caesar, and saying that he himself is Christ, a king.”
Who would have insured Jesus died so as to take the sins of the world upon Himself on the last sin altar? Were it not for Judas, the elders of the temple, Pilate, the people who rallied for Barabbas to be freed rather than Jesus? How else would Jesus purpose been accomplished?
As an aside, did you know that in Islam they teach that it was not Isa(Jesus) who was executed on the cross but Judas?