The only Jewish writings that exist from 1st century Judea (outside the New Testament, and maybe some apocalyptic writings that could be from the 1st century, but shed no light on it) are from Josephus, who does mention Jesus, John the Baptist, and James the brother of Jesus in three different passages, though the first has obviously had some filing at the hand of a later Christian scribe.
Doubting the existence of Jesus is a fringe indulgence though, since it is a far more complicated and improbable explanation for the creation of Christianity than accepting the basic story that there was an itinerant preacher who lived for a few years in Galilee and taught many memorable things. Somebody had to be responsible for the basic core of Christian teachings embodied in the Gospels, and if it wasn't Jesus then it doesn't really answer any questions to say it was somebody else.
In any case, it's unlikely someone creating it out of whole cloth would have included potentially damaging stuff like Jesus crying out on the cross, "My God, my God, why have you forsaken me?" That requires commentary (i.e., explaining it away) if you're a missionary trying to use the Gospel to convince a pagan that this fellow is the Son of God, as hammered down elsewhere in the book. Scholars call it the "Criterion of Embarassment" --- if something is included in the Gospel that's potentially embarassing and would require a long explanation to make it sound right, it's probably there because it was so widely known to be true that you could not omit it without losing your credibility. The cri de coeur is probably historical, then, because no fervent Christian ever would have come up with it otherwise.
The amount of accredited scholars, liberal or conservative, highly skeptical or otherwise, who believe Jesus did not exist is probably less than one-half of 1%.
Unfortunately, the Josephus passage is obviously a later editorial insert. Josephus' narrative stops right before the insertion and picks up on the other side; if the insertion were removed it would restore the narrative flow -- a good indication the piece was inserted. As the earliest extent examples of Josephus' work do not include the passage in question it's obvious a later editor inserted it.
As for your assertion there no other Jewish writings from the 1st century, a quick Google search shows that to not be the case. And as for establishing Christianity, that can be laid solely at the feet of Paul; the founding documents need be no more than something he and his circle cooked up either through a genuine belief that it was all divinely inspired, or as a cynical effort to make a little money.
"My GOD, my GOD, why hast thou forsaken me?", Christ's prayer and plea on the cross. It was recently taught to me that this plea, taken from Psalms 22 was actually a means of teaching that Jesus used.
Written 1,000 years before Jesus that crucifixion Psalms tells of His being despised, of things He would say and that would be said, about His bones being out of joint when His arms were pulled from socket on the cross, His thirst, lots being cast for His garments and his hands and feet being pierced. The 8th verse tells what the words of the High Priest were at the crucifixion (Matt.27:41).
To me, instead of trying to explain or understand why He would say He had been forsaken it makes sense that He was teaching us that prophecy was being fulfilled, from 1,000 years before His time.
And these people DARE to say there was no Christ.