There is compelling historical evidence for the existence of Jesus outside of the Bible. Among the extra-biblical references to Jesus, Cornelius Tacitus provides the most complete description supporting the biblical accounts of Jesus' crucifixion in his Annals 15:44.
Regarding Nero's mass persecution of Christian's in 64 A.D., Tacitus wrote a description of mass Christian persecution. This includes reference to Jesus (Christus, a latin transliteration of Christ, the Greek word for Messiah), being crucified (the extreme penalty) by Pontius Pilate. It also references a "most mischievous superstition" believed by Christians. This passage from Tacitus' Annals is considered authentic by historians. It is noteworthy that it portrays Christians negatively, making it an unlikely forgery of pious Christians. By 64 A.D. Christianity had expanded through the Roman Empire to such a degree that Christians became easy scapegoats for the great fire of Rome and became the brunt of mass persecution.
Nero fastened the guilt of starting the blaze and inflicted the most exquisite tortures on a class hated for their abominations, called Christians [Chrestians] by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judaea, the first source of the evil, but even in Rome, where all things hideous and shameful from every part of the world find their centre and become popular. (Tacitus, Annals 15:44)
The "mischievous superstition" Tacitus references is most likely the belief of Christians that Jesus returned to physical life again after dying on a cross. This belief was called "evil" by Tacitus. It appears to have engendered Roman hatred for Christians, and was something the Romans sought to restrain.
The early Christians' relentless adherence to the belief that Jesus was confirmed to be the Hebrew Messiah by returning to physical life again after brutal torture and crucifixion gave rise to Christianity as a major world religion.
Historian Will Durant reviewed the historicity of Jesus in his book Jesus and Caesar, a history of Roman civilization and of Christianity from their beginnings to 325 A.D. MJF Books 1971.
....That a few simple men should in one generation have invented so powerful and appealing a personality, so lofty an ethic and so inspiring a vision of human brotherhood, would be a miracle for more incredible than any recorded in the Gospels. After two centuries of Higher Criticism the outline of the life, character, and teaching of Christ, remain reasonably clear, and constitute the most fascinating feature in the history of Western Man. (p.557)
To suggest that Jesus was purely a mythological character is an unsupportable position.
For additional resources on the historicity of Jesus, see Recommended Reading.