I'm watching an episode of Doctor Who and it shows a few Roman soldiers that are played by black actors. At first I thought, "are they even trying to have any accuracy at all" but then I realized that Rome had territory in North Africa so I suppose someone from there could have volunteered for the Legion. Are there known cases of black men fighting for Rome?
What do you mean by 'black people'? If you include super-Saharan countries as 'black,' then the Romans are well-documented as having used Numidian cavalry and other troops from their African provinces. If you strictly mean sub-Saharan people, then there may have been a couple Ethiopians (cf. 'the life of Septimius Severus,' Historia Augusta), but they would have been very rare.