Copy/paste from here:

There is a small number of contemporary extrabiblical references to Jesus written, namely the account of Flavius Josephus:

About this time there lived Jesus, a wise man, if indeed one ought to call him a man. For he was one who performed surprising deeds and was a teacher of such people as accept the truth gladly. He won over many Jews and many of the Greeks. He was the Christ. And when, upon the accusation of the principal men among us, Pilate had condemned him to a cross, those who had first come to love him did not cease. He appeared to them spending a third day restored to life, for the prophets of God had foretold these things and a thousand other marvels about him. And the tribe of the Christians, so called after him, has still to this day not disappeared.

Josephus, Antiquities of the Jews

This account, though, it highly questionable, as it was written by Josephus, a first-century Jew, who would have viewed Jesus as a heretic rather than a "wise man". It was likely the case of Christian copyists later on editing his original less-than-flattering account (which likely no longer exists) but that's largely guesswork.

Tacitus also wrote of Jesus, referencing the burning of Rome:

Consequently, to get rid of the report, Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians 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 Judæa, 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. Accordingly, an arrest was first made of all who pleaded guilty; then, upon their information, an immense multitude was convicted, not so much of the crime of firing the city, as of hatred against mankind.

Tacitus, Annals

However, the question of whether Jesus exists is answered by one simple fact: Christianity's claims to Jesus' existence, never mind His divinity, began far too early for Jesus to be a complete fabrication. It's understandable if the first worshipers of Christ started appearing one or two hundred years after the supposed events, but Christians began appearing almost immediately, far too early for people to be making those types of claim, which others could easily refute.

"Jesus was crucified in Jerusalem during Passover week twelve years ago."

"No he wasn't, I lived in Jerusalem at that time and no such thing happened."

Something like that.

While people can certainly argue over the truth behind the biblical account, it wouldn't make any sense for there to never have been a Jesus at all. You can't just make up a person who never existed who supposedly lived such a short time ago, and there would be no reason to do so.

Remember that the first generations of Christians were persecuted and killed for their beliefs. It's understandable for someone who believes a false religion to die for it, not so much for someone who starts a false religion, and knows it's false, to die for it. The first Christians at least believed what they were claiming about Jesus.

Specifically to your question:

Note that to most of the Roman Empire, Jesus and Christianity weren't anything to 'write home about' until a couple decades after the fact. You're not going to have any Romans writing about Jesus during His ministry because Romans didn't care about Jesus during His ministry. It was when Christianity gained enough influence to actually be noticed that you can expect a written record of it.