18:28 Then they brought Jesus from Caiaphas to the Roman governor’s residence. (Now it was very early morning.) They did not go into the governor’s residence so they would not be ceremonially defiled, but could eat the Passover meal. 18:29 So Pilate came outside to them and said, “What accusation do you bring against this man?” 18:30 They replied, “If this man were not a criminal, we would not have handed him over to you.”
 

18:31 Pilate told them, “Take him yourselves and pass judgment on him according to your own law!” The Jewish leaders replied, “We cannot legally put anyone to death.” 18:32 (This happened to fulfill the word Jesus had spoken when he indicated what kind of death he was going to die.)

18:28 Then they brought Jesus from Caiaphas to the Roman governor’s residence. (Now it was very early morning.)

So after Annas questions Jesus and Peter's denial really late at night they take Jesus to the where the Roman governor was staying. Usually Pilate would be in Caesarea Maratima, off by the coast, but during big festivals he would go to Jerusalem to oversee things. Passover celebrates the Exodus from their Egyptian oppressors. This didn't breed kind feelings to their Roman overlords. John notes it's really early in the morning.

This is the site of the Muslim Umariyya Boys School, on the right, which was built over the Antonia fortress. Antonia was traditionally believed to be where Jesus was questioned by Pilate and the road in the photo is the beginning of the Via Dolorosa.

Historians and scholars, however, believe that Pilate was most likely staying at Herod's palace.

This is a model of Jerusalem around that time. The gargantuan complex up top is the Temple Mount, after Herod's expansion. Antonia Fortress is to the middle left while the towers that are part of Herod's Palace complex are on the lower right.

No, that's not a mega-giant up there, this is, again, a model. Herod's Palace complex is at the bottom of this image. This is most likely where Jesus was taken.

They did not go into the governor’s residence so they would not be ceremonially defiled, but could eat the Passover meal. 

The Jews stayed outside of the palace to keep clean. Here's Borchert's explanation on Passover cleanliness:

Although the laws of clean and unclean in respect to eating the Passover were complex, it seems that entering the residence of a Gentile would have been a major problem and would likely have rendered a Jew unclean for at least seven days and required the postponement of eating Passover for a month. The basic logic seems to have grown out of an interpretation of the rule of contamination from the dead in Num 19:11–13. It was widely believed that Gentiles aborted babies in their homes and either buried them within their homes or ran them down through their sewers. The uncleanness here was hardly the usual uncleanness of public encounter that could by sunset have been removed through a regular lustration or bath as an appropriate purification rite (cf. Lev 15:5–11).

So they didn't want to go in for fear of being unclean and so couldn't participate in the festivities. 

18:29 So Pilate came outside to them and said, “What accusation do you bring against this man” 18:30 They replied, “If this man were not a criminal, we would not have handed him over to you.”

Pilate goes outside to meet with this group. Since they're bringing a prisoner for Pilate to deal with, they have to have a charge against Him, so Pilate asks. They simply state that Jesus is a criminal, or else they wouldn't be there.

18:31 Pilate told them, “Take him yourselves and pass judgment on him according to your own law!” The Jewish leaders replied, “We cannot legally put anyone to death.”

Pilate tells them to deal with Jesus themselves, then. The Jews could deal with a lot of their own criminal situations with their own rules. But this case isn't a petty crime, but a serious one. They want to have Jesus executed, which they could not do. Only the Roman government could officially execute someone. There were mob stonings, such as the case of Stephen in Acts, but they were exceptions.

18:32 (This happened to fulfill the word Jesus had spoken when he indicated what kind of death he was going to die.)

Jesus had anticipated being lifted up which fits with a crucifixion, being lifted up on a cross, as opposed to the standard Jewish method of stoning.

It is ironic and tragic that those religious leaders stood outside of a Gentile place to maintain ritual cleanliness while in the pursuit of having someone wrongfully murdered. All of Jesus' words against them had hit their mark and they knew the truth. They followed their expansion of God's words but ironically not the manifestation of the Word of God, namely Jesus. They stuck to following certain rules while blatantly went against the commandment against murdering.

As Christians, we no doubt do similar things. What things are done at the expense of more important things? For example, we are often quick to wrongfully condemn others and call them out, but forget Jesus' commandment to love one another. We sing about how much we love Jesus in church to "be" Christian, but during the week sing songs about how much we love the world instead. Let's not by hypocrites like those that captured Jesus and had Him executed.