There can be made arguments for Jesus (if he existed or not) of being of almost any race.
He could be arab, such as the peoples who live in that area now.
He could be Jewish.
He could've been Egyptian. Slightly darker skin than Jewish.
He could have been black. The Nubians from farther south in Africa had already invaded Egypt, but at this time had also bee driven back down, so this one is very unlikey.
He could have been white (European). The Greek's had conquered all the way to India by 200 B.C. Many Greeks lived in this area because the mid-east was part of the Greek empire. Also the Romans took over most of the Greek empire, and were in control by now, but Greeks had lived here for hundreds of years. At the time Jesus existed, the Ptolemies ruled Egypt, they were Greek in ancestry.
He could have been a mix. Alexander the Great (greatest Greek conqueror) believed in a concept called Homonoia, where all cultures blended. He pushed this concept in his empire and encouraged inter-breeding and marriage among separate cultures. Also technically Jesus could have been of a race that has since been bred out of existence due to this.
Most likely he was Jewish in appearance. Since the majority of the population was Jewish in this area.
Also a little trivia for many. The Romans did not consider Jesus a criminal. The reason Jesus was executed was political. He had gained many followers (much like Buddha had hundreds of years prior). There was a growing tention between his followers and the majority population of people who didn't like a man saying he was the son of God. (Also the religious concept of a 'holy redeemer' was very popular around this tim, Christianity was only one of many religions that preached nearly the exact same teachings, and the other relgions all believed in their own 'holy redeemer', but it wasn't Jesus) Jesus was executed to prevent the mounting tentions from breaking out, he was executed to prevent a chaos and a possible civil war. The major reason that Christianity became popular was because of the work of one of the major writers of the Bible, Paul. He was converted to Christianity in his 20's about 50 years after the death of Jesus. He is mostly responsible for the spreading of Christianity.
And for those nitpickers, since I don't have my book with me while I write this, there may be some minor errors, but none that change what I wrote in any significant way.