It's impossible to prove a negative. You can't prove that unicorns don't exist. However, it's reasonable to assume they don't until evidence that they do exist surfaces. Likewise with Jesus. Until we have proof of his existence, assuming that he didn't exist is entirely reasonable.
There are some who claim that the burden of proof should be on those who claim Jesus didn't exist. After all, it's the minority opinion. The majority of scholars believe Jesus existed, so they're right. The problem with this line of thinking is that truth isn't a democracy. There was a time when the majority of people believed the sun revolved around the earth. The popularity of an idea has nothing to do with how true it is. If we really want the truth, we need to examine the evidence. Simply dismissing a theory because it isn't popular is counterproductive.
Another common misconception is that we should assume Jesus existed because of Occam's Razor. It's simpler to say that Jesus existed than to say he was a hybrid Greek/Jewish god who had a biography composed of Old Testament prophecies and a list of Cynic sayings attached to him. However Occam's Razor doesn't state that the simplest explanation is the right one, it states that the simplest explanation that fits all the evidence is most likely the correct one. Stating that Jesus existed simply doesn't fit the evidence as well as saying he didn't exist (see previous posts).
Some try to discredit those who present evidence that Jesus didn't exist by claiming they have an agenda. An agenda is certainly at work behind recent movies such as Religulous, The God Who Wasn't There, and Zeitgeist. I find it unfortunate that when most people think of the Christ Myth theory, they think of these movies which present a highly simplified and oftentimes counterfactual version of the theory. It's easy to dismiss the theory the way these movies present it.
However, scholars who advocate the Christ Myth theory don't believe that similarities between Christ and other deities alone is evidence for his non-existence. Serious scholars don't think that Christianity was a Roman conspiracy or that it was invented from whole cloth by Paul. It was a gradual process which evolved over hundreds of years. Besides, even if mythicists do have an agenda, it doesn't automatically mean that they're wrong.
Why do scholars doubt the existence of Jesus, but not other historical figures? Actually, some scholars do doubt the existence of other historical figures such as Confucius, Muhammad, Aesop, and William Tell. Some scholars even doubt the existence of Paul and John the Baptist. Socrates is another figure which some scholars think may not have existed, but there is more evidence for his existence than for Jesus. (See http://www.dougshaver.com/christ/socrates/socrates.html.) No one questions the motives of those who doubt Socrates' existence since scholarly curiosity is motive enough.
I could just as easily be writing a blog doubting the existence of Lao Tze, the alleged originator of Taoism. However, I choose to write about the existence of Jesus because I was raised in a predominantly Christian culture and I find the question of Christian origins fascinating. I have no axe to grind, I'm simply interested in the subject. Besides, even if I did have an axe to grind, that wouldn't invalidate the evidence. Shooting the messenger won't change the message.
Except for fundamentalists who believe everything in the Bible is literally true, everyone agrees that the Jesus portrayed in scripture is at least partly mythical. Scholars in search of an historical Jesus use a set of methods to separate the man from the myth. Miracles such as walking on water likely didn't happen, however, could there have been a man named Jesus who was crucified by the Romans? Since Jesus was a common name and the Romans crucified often, there were no doubt numerous crucified Jesuses, but was one or more of them the basis for the Jesus of scripture?
I don't think it's possible to know for sure one way or the other. However, some scholars think there is evidence for an historical Jesus, so let's examine some of the methods they use to look for him.
Criterion of cultural congruency: A source is more credible if it fits the culture context.
This criterion can help us rule out a lot of sources, but it doesn't help us determine if Jesus really existed.
Criterion of linguistics: Since Jesus spoke Aramaic and the Gospels were written in Greek, anything that only makes sense in Greek wasn't spoken by Jesus.
Again, this is helpful to rule out sources, but not rule anything in. However, as an aside, there was a tremendous amount of Greek influence in Israel before, during, and after Jesus' time, so it's not impossible for Jesus to have spoken Greek.
Criterion of ancientess: The older the source, the more reliable it is.
Fair enough. However, most scholars assume Jesus lived during Pilate's time and this hasn't fully been established. Perhaps the historical Jesus lived further back in history than has popularly been assumed. For example, many scholars assume that Christianity predated Gnosticism, but it could just as easily be the other way around. Christianity could have evolved out of Gnosticism.
G. R. S. Mead presents compelling evidence that Jesus may have lived in 100 BC. The mingling between Jewish and Greek cultures goes back hundreds of years before that. The anachronisms in the Gospels (such as the presence of Pharisees in Jerusalem before the destruction of the temple, Jesus being born and raised in Bethlehem and Nazareth before those cities were founded, and Jesus referring to the murder of Zechariah in Matthew 23:35 long before it happened) prove they were written long after the time of Pilate. Paul provides us with scant details about the life of Jesus. For all Paul tells us, Jesus could have lived earlier than 100 BC. If the Jesus of the Gospels was based on someone who lived hundreds of years before and of whom almost nothing was known, then speaking of an historical Jesus at all is meaningless.
Criterion of embarrassment: Anything that would have been embarrassing to the early Christians must have really happened, otherwise they wouldn't have mentioned it.
The problem with this criterion is that we can only guess what would have been embarrassing to the early Christians. The criterion of embarrassment is often used to prove that Jesus was crucified as a criminal because early Christians would have been embarrassed to admit that. Why would the crucifixion of their leader be embarrassing? Mormons aren't embarrassed by the fact that Joseph Smith was shot to death in jail. Having a leader considered an outlaw by the authorities is far from embarrassing if you consider the authorities evil and your leader wrongfully executed.
Criterion of attestation: If two or more independent sources state the same thing, the event must predate both sources.
This is tricky, because you have to determine what is considered an independent source. Matthew and Luke were both based on Mark, so if all three say the same thing, it doesn't count as three separate witnesses, just one. Even the Gospel of John shows an awareness of the synoptics, so that can't be counted. Historical sources, such as the references to Jesus in Josephus, are questionable at best. The epistles give us no biographical information except the crucifixion, and that's also questionable. Q is often cited as an independent source, but Q is a hypothetical document and much of Q is based on Cynic writings which go back centuries. We have to find independent sources first before this criterion is useful.
Author's agenda: If something fits the author's perceived agenda, it is suspect.
This criterion can be used to rule out pretty much everything in the Gospels. The writers of the Gospels wanted Jesus to fulfill every prophecy in the Old Testament, therefore any time he does fulfill a prophecy, we should assume the author made it up. The Gospel of Mark reads like someone went through the Old Testament and combined all the prophecies together into a narrative story. When you rule out everything with an Old Testament precursor, you're not left with much besides Gnosticism and Cynic philosophy.
Could the historical Jesus have been a Gnostic or a Cynic philospher? Several scholars have advocated just that. However if we're left with no biographical information to speak of, the historical Jesus could just as easily be the collected sayings of an entire community. Proposing that all of the sayings come from a single person is unnecessary and unprovable.
Criterion of difficulty: If a character is invented, material that creates difficulty for the narration will be left out.
It's been argued that Jesus being born in Bethlehem, but raised in Nazareth fits the criterion of difficulty. Both Matthew and Luke create absurd stories to explain why Jesus was born in Bethlehem, but raised in Nazareth. According to Luke, Joseph and Mary lived in Nazareth, but had to go to Bethlehem for a highly unusual census. According to Matthew, Joseph and Mary lived in Bethlehem but had to flee to Nazareth to avoid the slaughter of innocents. This account contradicts Mark and John which state that Jesus was born in Galilee where Nazareth is located (Mark 6:1, John 7:41-43).
Therefore, Jesus wasn't born in Bethlehem, but Matthew and Luke invented stories to say he was so that he would fulfill the prophecy of Micah 5:2. Jesus being from Nazareth creates difficulty for the narrative, therefor Jesus of Nazareth was a real historical person. If Jesus had been invented, the authors would just say he was from Bethlehem to begin with and avoid any reference to Nazareth.
This criterion seems reasonable at first glance, but upon closer inspection, it's revealed to be quite silly. Consider Matthew 21:1-7 where Jesus rides both a donkey and her colt into Jerusalem at the same time. If we apply the criterion of difficulty, we'd be forced to say that Jesus really did ride upon two donkeys at once because this rodeo stunt creates serious difficulty for the narrative.
However, things make a lot more sense if we ditch the criterion of difficulty and instead say Matthew misunderstood the poetic repetition of Zechariah 9:9: "Behold, thy King cometh unto thee: he is just, and having salvation; lowly, and riding upon an ass, and upon a colt the foal of an ass."
As for Jesus being from both Nazareth and Bethlehem, that's easily explained the same way. Matthew and Luke are trying to fulfill two contradictory prophecies: Micah 5:2 and the one referred to in Matthew 2:23 which says the savior will be a Nazorean. Difficulties in the narrative don't have to be caused by trying to fit a real person into prophecy when they can be explained more easily as an attempt to accommodate two contradictory prophecies.