Jesus and Pilate spoke in Greek. What was spoken was not “proper” Greek, as Josephus described it, but the local version, which included a few Aramaic words. Almost all Christ’s conversations in the NT were in Greek. The few exceptions are noticeable because often the Aramaic is quoted and then translated into Greek for the reader, who was assumed to speak Greek.
Greek was the common language of the times, spoken through all of the eastern and southern parts of the Roman empire. All educated Romans were expected to know and speak Greek as well as Latin.
Jesus also spoke the local Aramaic, but the simplest argument for Jesus having taught in Greek is the occurrence of untranslated Aramaic word and phrases in the Greek of the Gospels. Many of these words are not translated in English versions of the Bible either. They can appear as single words such as "mammon", or as complete phrases such as "Talitha cumi" or "Eli, Eli, lama sabachthani?". In the latter two cases, not only to the untranslated Aramaic phrases appear, but they are followed by a translation in the Greek of the Gospels.
The local Greek included a few Aramaic words. One of Christ's catchphrases, translated in the KJV as "verily I say to you," contains an Aramaic word, "amen", which is translated into "verily". Of course, we are now familiar with this word because it has made its way into the English language for a special purpose, the ending of a prayer. There is a Greek word that has a very similar meaning, men, that also means "truly." Why not use it? Because Christ didn't speak proper Greek. He mixing a little Aramaic with the Greek. However, I don’t see any such words spoken to Pilate, a Roman who may or may not have understood them.
Many scholars today claim that Jesus spoke Aramaic and the Greek of the Gospels was a translation from Aramaic. These claims appear to be made largely for political reasons. The scholars involved are usually Jewish or Muslim, both of which speak Aramaic languages. If the Bible was a translation from Aramaic, why not translate the words and phrases above? How would the supposed Aramaic translator pick which Aramaic words not to translate? It makes no sense.
Aramaic also makes no sense because Jesus’s words in the Gospels include many, many plays on words and even rhymes that only work in Greek. The translator would have had to add all of this as part of the translation. This again makes no sense.
For a lot more on this topic, you may want to read this article about the evidence that Jesus spoke Greek.