Why did the Jewish high priests deny Jesus if He was fulfilling their scriptures?
It seems as if they viewed Him as nothing to do with their God.
What are some reasons from the perspective of the high priests that they refused to accept Jesus?
The Jewish priests expected the Messiah to come in like a Lion with great fanfare and overwhelming military might for subduing the enemies of Israel. They did not expect the Messiah to be born in humble circumstances or to grow up as a common craftsman.
Jesus threatened the authority and the lifestyle that the priests had grown accustomed to, as they grew very rich while helping the Roman authorities to maintain control over Israelite-controlles towns. Several times in the Gospels (Mat 23:1-4, Luke 11:37-44), Jesus directly rebukes the priests on multiple issues.
Many people were putting their trust in the teachings of Jesus instead of the priests.
It must have been difficult for the Jewish Priests to consider that the promised Messiah would come as a threat to themselves, God's representatives to His own people. They viewed him as being opposed to the established authority that controlled the Temple and the People, which would likely seem contrary to what they might expect from the long-awaited Messiah.
The account is detailed in Matthew 26 starting verse 57.
First, the Sanhedrin attempted to convict Jesus on false pretenses, but that failed. Then this was said:
In verse 64, Jesus basically calls himself God. That is blasphemy.
Any prophecies that Jesus might have fulfilled was apparently irrelevant to the Sanhedrin. Continuously in the Gospels, they attempt to trap Jesus, trick Him into sin that warranted arrest, and they even went to stone Him for blasphemy earlier (See John 10:22). They just plain didn't like Him, mostly because He did not fit in the box they wanted the Messiah to fit in, which was more of a conqueror that would bring Israel back to its glory days of King Solomon. They also did not expect the Messiah to claim to be God. Further, Jesus continuously aggravated the Sanhedrin by making them look like fools and hypocrites.
To be sure, the Jewish leaders should have accepted Jesus. The miraculous signs He performed was irrefutable evidence that He had come from God, as Nicodemus, himself a Pharisee, noted.
However, God has prophesied this through Isaiah the prophet, as John notes later.
However, many of the rulers, including Nicodemus and Joseph of Arimathea, did, in fact, believe in Jesus and accept Him as their Messiah. Yet, the Pharisees as a group were imposing significant consequences on anyone who followed Jesus, so there was a lot of fear that led to many believing in secret. Sadly, it seems, at least prior to Jesus' death and resurrection, that they loved the approval of men more than the approval of God.
The high priests didn't care about the scriptures. The high priests at that time were agents of Rome.
It was standard procedure for the Romans, after conquering a nation, to depose the nation's leaders (both political and religious) and replace them with people who would be loyal to Rome.
In the case of Judea, that meant replacing the Hasmonean king with Herod, and replacing the Levitical high priest with Annas ben Seth. (This is the same Annas mentioned in the gospels as being high priest along Caiaphas. Technically, Annas had been appointed by Quirinius--the same Quirinius of the famous census--and deposed by Valerius Gratus, Pilate's immediate predecessor. However, Annas was succeeded first by his son Eleazar and then by his son-in-law Caiaphas, and following Caiaphas, by four more of Annas' sons.) During the years of Roman rule, Annas and his family held the title of high priest for more than half a century. Caiaphas himself managed to hold the office for 18 years, including Pilate's entire term as governor.
In other words, this was a family that had earned the Romans' trust, and they were rewarded for it.
That's why, when Jesus showed up with followers proclaiming him to be the Messiah, the high priests saw him as a threat. As Jeff Wolski points out in his answer, most Jews at the time expected the Messiah to free them from Roman rule. This would have meant a return of the high priesthood to the Levites, and a loss of power for the family of Annas. And that's why they wanted him dead.
Caiaphas thought that Jesus' death would appease Roman leaders who perceived Jews as unruly, ungrateful subjects. He didn't even realize his words would have a deeper meaning.