Christianity is a historical faith. Christianity is founded on the figure of Jesus and so it really matters whether Jesus was a real person or not. If he did not exist then our seeking ends there. But if Jesus did exist then we can move on to the more interesting questions of what he did, who he claimed to be, and in what ways he is special.
What is unusual about the question of Jesus’ existence is that it is a loaded question. The majority of historians recognise that Jesus existed and yet you will find plenty of sceptics who think that the existence of Jesus is still an open question. Of all known individuals from the first century AD very few feature in as many sources written within living memory as Jesus. Yet of all known individuals from the first century AD it is Jesus whose existence is doubted. This disparity between what reliable historical methods indicate and what sceptics choose to doubt tells you something about the sort of question this is.
So let’s recap the evidence, starting with non-Christian sources. Perhaps the most well-known source is Josephus, who in one passage talks about "Jesus … a doer of wonderful works", whom Pilate condemned to the cross (Antiquities 18:63-64). Many people think this text has been tampered with because it includes the line "he was the Christ" (and it's unlikely that Josephus, a Jew, would have said that) so we cannot be sure what this passage originally wrote. But this is not the only time Josephus mentions Jesus. In another passage he talks about an early Christian elder, who was stoned to death; Josephus describes him as "the brother of Jesus, who was called Christ" (Antiquities 20:9).
Another Jewish writer, Mara Bar-Serapion, mentions in a letter to his son a "wise king" whom the Jews put to death. Many scholars believe that this "wise king" was Jesus. The Talmud, the book of Jewish tradition, also contains stories about Jesus. Though greatly embellished and particularly anti-Christian, the Talmud refers explicitly to the death of Jesus "on the eve of the Passover" (Sanhedrin 43a).
Jesus is also mentioned by two Roman historians. Tacitus records that "Christus … suffered the extreme penalty" under Pontius Pilate (Annals 15:44) and Suetonius mentions one "Chrestus", whose followers were causing disruption in Rome (Life of Claudius 25). The Roman satirist Lucian writes about the founder of Christianity, who was crucified (The Death of Peregrine).
Early critics of Christianity, like the Platonist philosopher Celsus, did not dispute that Jesus existed – they only disputed the claims he made.
Of course, our main source of information about Jesus comes from texts written by Christians in the first century. These include texts from the late 40s or early 50s, such as James and the Didache; texts from the mid-50s, such as Paul’s letters to the Corinthians, Thessalonians and Romans; and, of course, the gospels, which probably date from the 60s or 70s. All easily within living memory of the events of Jesus' life.
This is why historians do not doubt the existence of Jesus – you'd need to ignore all the evidence.