There is very little question that Jesus the human being and Roman subject existed, professed some sort of fanatical messianic religious messages, and got executed for it. Stronger than any evidence for Jesus directly is the evidence for the existence of his brother, James, who was a respected religious leader in Jerusalem for about 25 years after the death of Jesus. It is James that Josephus was writing about--Jesus is only mentioned to explain who James is, "the brother of Jesus who some call Christ." But in addition to this one near-contemporaneous reference, there is also a dozen or so writers who talk about James in the 2nd century. That is a century after his death and the death of his brother, but these writers seem to be working from largely different sources so we can infer multiple historical accounts that would have existed at the time. If we agree that James existed, then the strongest argument for the existence of Jesus is that all of these motivated, non-Christian secular writers of the 1st and 2nd century would have been in a better position to know whether Jesus had existed, and if there was any doubt they would have definitely smeared James as having made up his imaginary brother. But none of them did.
If anyone is interested in this question btw, Reza Aslan's Zealot is an approachable book on the historical Jesus, what little we know for certain and the surprising amount we can confidently infer from the available evidence.