It cannot be known, with absolute certainty, whether Jesus was black or not. This is because the Bible gives us almost no physical description of him.
The lineage of David
What we do know is that Jesus was a Jew and a descendant from the tribe of Judah. He was specifically a descendant of David, King of Israel, who was described by the prophet Samuel as not being black or dark skinned but looking "ruddy," meaning red or reddish color.
And Samuel said to Jesse (David's father and a person in the lineage of Jesus), 'Are all the young men here?' Then he said, 'There remains yet the youngest . . .'
. . . and Samuel said to Jesse, 'Send and bring him . . . ' Now he (David) was ruddy (Hebrew word admoniy, Strong's Concordance Number #H132, meaning reddish of the hair or complexion), with bright eyes, and good-looking (1Samuel 16:11 - 12).
Note also that this ancestor of Jesus had bright eyes and was good-looking, although having beautiful ancestors is certainly no guarantee of how a person looks.
Isaiah's description
Isaiah the prophet, although he does not mention the Messiah being any particular skin color like black, does give us a general description of his appearance.
It was the will of the Lord that his servant (Jesus) grow like a plant taking root in dry ground. He had no dignity or beauty to make us take notice of him. There was nothing attractive about him, nothing that would draw us to him (Isaiah 53:2)
We also know, based on the Biblical principle that men in general should not have long hair, that Jesus had his short (see 1Corinthians 11:14, 16). This is one (of many) reasons why the image of a man on the Shroud of Turin is decidedly NOT that of Christ!
One former well-known religious author and preacher wrote extensively concerning what he thought Jesus looked like. He believed he was of average height for his time. Additionally, because of the hard work of helping his stepfather Joseph in the carpentry business, the author thought the Lord was likely thinner rather than fat and was somewhat physically strong due to the activities required of such an occupation.
Average looks
Our Savior, almost certainly, looked like the average Jew who lived in Palestine during the first century A.D. The purpose of his life was not to draw attention to himself by his looks but rather promote the worship of the true God through the gospel he taught.
Considering the Biblical description and racial characteristics given us by secular history, the evidence points to the fact that Jesus was not black or African in ancestry or looks. The important thing is that, no matter what he looked like or what color was His skin, He was (and is) the Son of God who willingly gave his life for ours.