The bible doesn't really get into the business of assigning blame for christ's death, either to the Jews or to anyone else. Within the broader narrative, Christ's death was not really something that could have been avoided, especially if, as most Christian do, you read the entirety of the old testament sacrificial system, poetry, and archetypes as all pointing forward to his death as something of significance.

The gospel writers themselves record several instances of Christ predicting his own death, both to his disciples and to others (in fact they record some of these predictions being used out of context as evidence at his trial).

All in all, this does not really create a narrative where the assignation of blame is really called for, and in my reading of the texts, I don't see it. What I do see is the following:

First, there is a lot of effort made to make it clear that Jesus was innocent of the charges, and that they were only made to stick through a gross miscarriage of justice. Only select members of the sanhedrin were called, it was carried out at an illegal hour, at an illegal speed, using lying witnesses, in the service of an administration that had other motives (namely, the protection of their power). These motives might be obvious, but in the narrative, this serves primarily to emphasize Christ's innocence rather than to lay blame at the feet of the leadership (though they definitely were guilty of a miscarriage of justice).

Second - in the case of Pilate, the narrative emphasizes his inaction and unwillingness to do the right thing under pressure - i.e. it explains why he didn't stop the crucifixion but instead authorized it, more or less painting him as a weak man unwilling to take a stand for what he knew to be right (in Matthew it even describes his wife coming to him to tell him that Jesus is innocent).

Basically, while there is plenty of potential for blame to go around, as you might expect in the deliberate execution of an innocent man for political gain by various factions of political elites, the narrative has other goals; making clear that Christ was innocent, showing that he was having a hard time (treated bad, rejected and abandoned by friends) and being pretty clear regarding how the events matched all the foreshadowing drawn from both various books of the Old Testament, and from earlier in the gospels. The one time when Jewish everyday "people" are tied in to the story, the bible is quite clear that the crowd was instigated by "chief priests" i.e. the political elite to whom Jesus is portrayed as a threat (Matt 15:11), and just to make it clear that blaming anyone is not his plan here, the gospels include the line "Father forgive them, for they know not what they do" (Luke 23:34).

Basically, while people have often assigned blame for the death of Christ, they do it based on their own motivation and definitely not on any real scriptural basis. The history of antisemitism whether in early Christian/Jewish tension, or in later tension following the reconquista or going right up through WWII, appears to be far more tied up with political upheavals, ethnic tension and fears brought on by social change than to a careful reading of scripture - though of course the bible has been used in the service of these movements, and these tensions have used religion as a rallying cry.

I really do suggest reading the the relevant sections of each gospel and looking as broadly as possible on the story - though bear in mind that within the larger arc of the gospel writers, these few chapters represent the finale and resolution not only of their own books, but of the entire bible, and as such are written in that context with a tonne of references to other books.

Matthew Chapter 27, verses 22-26

^(22) “What shall I do, then, with Jesus who is called the Messiah?” Pilate asked.

They all answered, “Crucify him!”

^(23) “Why? What crime has he committed?” asked Pilate.

But they shouted all the louder, “Crucify him!”

^(24) When Pilate saw that he was getting nowhere, but that instead an uproar was starting, he took water and washed his hands in front of the crowd. “I am innocent of this man’s blood,” he said. “It is your responsibility!”

^(25) All the people answered, “His blood is on us and on our children!”

^(26) Then he released Barabbas to them. But he had Jesus flogged, and handed him over to be crucified.

(NIV)

This account from the Gospel of Mark shows Pontius Pilate asking the crowd (They) what should be done with Jesus Christ.

It then describes Pilate's reaction to the crowd's demand of crucifixion. He had Christ whipped, then handed him over (presumably to the Sanhedrin, which was a council of Jewish Spiritual leaders[who brought the charge of blasphemy against Christ in Matthew 26, 57-67]) for crucifixion. Pilate also released Barrabas, who in the Gospel of Matthew, Chapter 26 verse 16 is described as a "notorious criminal."

It is true that Pilate relented only when the crowd refused to accept any other outcome that the death of Christ. It does not mean that he is innocent of murder. His symbolic washing of his hands (which symbolizes his turning a blind eye; and therefore claiming innocence) would hold no water with either modern or Roman law.* Would it not be great if the most-hated man in each city were brought to each mayor for their symbolic approval of a vigilante-justice public execution*?

Consider Pilate's position as governor of a Roman province. A governor is the penultimate authority, subordinate only to Rome, or by this time, the Emperor. Pilate was, in effect, the final authority on this matter of whom to release. Pilate had legions of soldiers under his command, he could have dealt with the consequences (Likely riots)of not condemning an innocent man (Christ) to death.

In all fairness, Pilate did not bring the charge of blasphemy against the Messiah, nor did he ask for the death penalty. The Jewish leadership holds the blame for that.

Therefore, the blame lies only on the Sanhedrin and Pontius Pilate for the death of Christ.

Note: Since you mentioned a Catholic upbringing, you may be interested to know that Pope Benedict XVI denounced blaming the murder of Christ on the Jewish people. So, if there was retroactive blaming of the Jewish people, the Papacy most assuredly disagrees with that position.

TL;DR

The Jewish Leadership, and Pontius Pilate both share the blame for Christ's murder. Nowhere in the gospels does it call out the Jewish race as a whole for the murder of Christ. The Jewish people have little to no blame in this most horrible and yet beautiful act of crucifixion.

Of course I submit my answer to moderators such as /u/husky54 for correction.