Many Christians make the mistake of believing that Song of Solomon was literally written by King Solomon. The book is actually an operetta and has parts for a female singer and a male singer. In verse 1:5-6, the female singer describes her skin as dark because, as a farm worker, she spends a lot of time in the sun. She asks the townspeople not to look down on her because of this - something you could not imagine a rich and powerful king saying.
He could have been, there are no pictures to prove as to what his skin colour was, we only know that he was Jewish,, so he could have been darker than most Caucasian people, or maybe lighter than the average Jewish person, most people from the Eastern European countries are dark skinned, it is also called Mediterranean, like southern Italian people are darker than Northerners.
Firstly, Solomon was a Jew, and not black as the question suggests. He came from a heritage when marrying outside the Jewish race was taboo, and therefore his ancestry would have been very Jewish and almost exclusively pure-bred.
Secondly, the verses cited in the Song of Solomon refer to the lover (whether it was Solomon or not) being dark and suntanned - not black. The original Hebrew confirms this as the words used suggest a 'tanned' appearance rather than the black skin of the African, who were well-known both in the times of Solomon and when the Jews were in exile in Egypt. The distinction in the Hebrew makes this very clear.
As in Victorian England, white, pale (or untanned skin) was seen in Jewish society as more attractive than tanned skin and was believed to be something to which one should aspire. The reason for this was that those who were tanned were often those who were the lowliest people - those who worked outside on the land in the worst jobs. Nobility, or royalty, who had no need to work, had no need to be outside on hot days, and so retained the olive-skinned look of the Jew.
This is confirmed in subsequent verses where the lover makes known that he is actually not a worker in a vineyard, though his skin is tanned, but the keeper of a vineyard - a much more lucrative profession and more attractive for the female as he was not a peasant but a man of substance.
Therefore, the moral is, never look up Biblical passages out of context. Never cite one verse as 'evidence' of something or as a mandate for your own prejudices lest you totally misinterpret what was originally intended.
Therefore, as Solomon wasn't black then neither was Jesus, who came, as did Solomon, from Jewish ancestry and would have been olive-skinned like all Jews at that time.