A recent episode of Doctor Who (a show not exactly known for it's historical accuracy) shows us Scotland in the 2nd Century AD, where soldiers in the Roman Legion are invading and fighting the Picts in the area. One of the soldiers is black. Is it historically feasible to have had a black soldier in the Roman army in that area at that time?
This wouldn't have been unusual at all (as a side note, the Emperor Septimus Severus who rule from AD 193 - 211 is believed to have been of African descent on his father's side)
As an archaeologist I'm less familiar with textual evidence (don't kill me, historians!) so I'm going to list what I'm familiar with;
There is the military tombstone at Birdoswald (on Hadrian's wall) of a gent called C. Cossutius Saturninus who was from Hippo Regius in what is now Algeria.
A site identified as a manufactory for military goods has types of pottery hailing from Africa as well as graffito in the Punic language
We also know of 1 Legate (Legionary commander) who was African and another two names which are probably African.
(This is from an Imperial Possession by David Mattingly, pgs 180-181)
The Roman army (and Roman culture in general), was pretty multicultural. A large proportion of the military was made up of what are called the Auxiliaries, which were troops who weren't citizens (you had to be a citizen to be a legionary at this point in time, and most people in the empire weren't citizens). However, after serving their time, the auxiliaries would be granted citizenship and sometimes given land to settle on, oftentimes in the area they had been serving in. So an African unit might be sent to Britain, spend their time fighting uppity Picts and then be given land to settle down in right there in Britain.
It would be very uncommon but not entirely impossible. The main roman legions stationed in Britain were mostly recruited from Italy, Spain and Roman parts of Germany. Occasionally you'd get soldiers coming from roman north Africa as Typologyguy says but most of the evidence points to people coming from places like modern Algeria and Libya. People living there would be a mix of Berbers, the natives of that region, people related to the Carthaginians or Italian colonists, none of which are Sub-Saharan black. Now Nubia which is modern day Sudan was part of the roman empire during some periods and the people there were mostly Sub-Saharan Africans so you would get the occasional Sub-Saharan African but they would be a hell of a long way from home by ancient standards even with Rome's large system of trade.