As the Carthaginians were North Africans/Syrians, I don't think racially they would have been "black" http://upload.wikimedia.org/wikipedia/commons/e/ef/Mommsen_p265.jpg

This is a bust of Hannibal (supposedly found in Capua , Italy, although I'm not sure if it's ever been verified as authentic). On that bust it doesn't seem like he holds typical "black" features. However, they probably would have had "black slaves" as noted in Herodotus " The Carthaginians also say they trade with a race of men who live in a part of Libya (Africa) beyond the Pillars of Hercules (Straits of Gibraltar)."

As noted, Carthaginians were Phoenician colonists of North Africa. So maybe they had Phoenician upper class with Berber lower class. Probably not the 'black' you are looking for.

I am just curious, why do you ask? Just find the question interesting :)