There are a handful of mentions of a Jesus or a Christ in Roman sources. All of them come from decades later, and none of them are particularly focused on Jesus. They instead tend to mention him in conjunction to Christians.

For example let's take a look at this passage from Tacitus' Annals.

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.

This was written around 116 AD. So it was likely written after all of the books in the New Testament. All it tells us about "Christus" is that he was executed by Pontius Pilate. Nothing about the nature of his death, his teachings, or his deeds. Even then it's questionable as to where Tacitus got his information on Christians. It's likely that this information came from Christians themselves, and so either came from the gospels or from the same source that the gospels drew from.

Other sources like Pliny the Younger's Letter to Trajan from around 112 AD do much the same. They confirm that roughly 80 years after his death, Jesus had a group of followers. As to the man itself it provides less than Tacitus did.

The last early mention comes from the Jewish Historian Josephus in his Antiquities of the Jews. Perhaps I should call this the first mention as it is actually the oldest mentions we have of Jesus. It was written around 93 AD.

so he [the High Priest Ananus] assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king, desiring him to send to Ananus that he should act so no more,

This is generally considered a reference to James the Just the purported author of the Epistle of James. It should be pointed out that some people (most notably Richard Carrier) think the whole "who was called Christ" parenthetical was a late insert and not part of the original text. They think the "Jesus" in the text refers to Jesus ben Damneus, who became High Priest right after Ananus. Neither Jesus (Yeshua) nor James (Ya'aqov) were uncommon names in first century Judea.

This is also a good time to mention that we have an ossuary (a box for holding bones, similar in function to a modern cremation urn) which has this inscription on it - Ya'akov bar-Yosef akhui diYeshua. Which translates as "James, son of Joseph, brother of Jesus." The age, providence, and authenticity of the ossuary are all disputed. But it does exist and may be evidence for existence of James, Joseph, and Jesus. But even if authentic, it might refer to a different group of men. Joseph isn't a rare name either.

Antiquities also has what is called the Testimonium Flavianum. But that is largely considered a forgery that was inserted sometime between the 2nd and 3rd century. It doesn't sound like Josephus, and largely uses language that was common centuries later. So I won't detail it to far.

And that's it. That's the sum total of third party early writings and evidence about Jesus. Not that we should be surprised. If he existed as described, he was a relatively unknown leader of an obscure Jewish sect in a time and place where there dozens of them. Such men don't generally get contemporary biographies.

/u/talondearg wrote a great response to a very similar question here. You might find it interesting.