First off, I do want to clarify that the Romans had no conception of 'race' as we know it - as summed up rather excellently (This can count as the TL;DR, if you like) by /u/BonSequitur in another thread...

Romans had their own ideas about classifying people based on their physical characteristics, cultural background, and parentage; it's not very rigorous to refer to those ideas using the word 'race.' A Roman for example wouldn't have grouped Gauls, Goths, and Latins together into a 'white race' that is binarily opposed to a Lybian 'black race,' for example.

However, the Romans had tons of interaction with Africans in general - whether or not they were "black" is unknown (Honestly, define black.), and they had different views of all of them. For example, the Numidians in particular were renowned as excellent horsemen, able to outperform any other riders with their incredible synergy with their horses. The Carthaginians were known far and wide (Before they were wiped out) as the greatest trading empire of the Mediterranean. One rather hilarious view of the Sub-Saharan Africans is noted by Pliny (Long quote, but totally worth the read, I promise!) :

After passing all these peoples, there are vast deserts towards the east until we come to the Garamantes, the Augylæ, and the Troglodytæ; the opinion of those being exceedingly well founded who place two Æthiopias beyond the deserts of Africa, and more particularly that expressed by Homer, who tells us that the Æthiopians are divided into two nations, those of the east and those of the west. The river Nigris has the same characteristics as the Nile; it produces the calamus, the papyrus, and just the same animals, and it rises at the same seasons of the year. Its source is between the Tarrælian Æthiopians and the Œcalicæ. Magium, the city of the latter people, has been placed by some writers amid the deserts, and, next to them the Atlantes; then the Ægipani, half men, half beasts, the Blemmyæ, the Gamphasantes, the Satyri, and the Himantopodes.

The Atlantes, if we believe what is said, have lost all characteristics of humanity; for there is no mode of distinguishing each other among them by names, and as they look upon the rising and the setting sun, they give utterance to direful imprecations against it, as being deadly to themselves and their lands; nor are they visited with dreams, like the rest of mortals. The Troglodytæ make excavations in the earth, which serve them for dwellings; the flesh of serpents is their food; they have no articulate voice, but only utter a kind of squeaking noise; and thus are they utterly destitute of all means of communication by language. The Garamantes have no institution of marriage among them, and live in promiscuous concubinage with their women. The Augylæ worship no deities but the gods of the infernal regions. The Gamphasantes, who go naked, and are unacquainted with war, hold no intercourse whatever with strangers. The Blemmyæ are said to have no heads, their mouths and eyes being seated in their breasts. The Satyri, beyond their figure, have nothing in common with the manners of the human race, and the form of the Ægipani is such as is commonly represented in paintings. The Himantopodes are a race of people with feet resembling thongs, upon which they move along by nature with a serpentine, crawling kind of gait. The Pharusii, descended from the ancient Persians, are said to have been the companions of Hercules when on his expedition to the Hesperides. Beyond the above, I have met with nothing relative to Africa worthy of mention.

Needless to say, while some of these may have been tall tales originally based on truth (towns in the desert could be a reference to desert nomads who live from oasis to oasis), it's not terribly true to fact. But, let's get down to the question that you asked regarding "black" people.

I suspect that the people that you might be most interested in would be the Ethiopians - whose country is described in depth by Pliny here. I won't quote that one TOO heavily, as it's even longer than the last quote, and Lord only knows that too many quotes get scary. However, he makes a few more fanciful notes, such as this one here, claiming that...

It is not at all surprising that towards the extremity of this region the men and animals assume a monstrous form, when we consider the changeableness and volubility of fire, the heat of which is the great agent in imparting various forms and shapes to bodies. Indeed, it is reported that in the interior, on the eastern side, there is a people that have no noses, the whole face presenting a plane surface; that others again are destitute of the upper lip, and others are without tongues. Others again, have the mouth grown together, and being destitute of nostrils, breathe through one passage only, imbibing their drink through it by means of the hollow stalk of the oat, which there grows spontaneously and supplies them with its grain for food. Some of these nations have to employ gestures by nodding the head and moving the limbs, instead of speech. Others again were unacquainted with the use of fire before the time of Ptolemy Lathyrus, king of Egypt. Some writers have also stated that there is a nation of Pygmies, which dwells among the marshes in which the river Nile takes its rise [...]

Pliny amuses me :P Anyways. Interestingly enough, the Romans enveloped the Ethiopians into their mythology - Pliny himself makes a reference to it, however, the best source for THIS matter that I'll turn to is Ovid's Metamorphoses - a brilliant work of poetry on Roman mythology. In it, he discusses how Phaethon attempting to control his father's chariot resulted in the skin of the Ethiopians charring and turning black - which they just happen to be to this day. (Line 235). The bit that interested me the most, though, was the story of Perseus and Andromeda. For those who don't know the tale (and who are too lazy to read the link), I'll TL;DR it real fast for you :)

Perseus, after killing Medusa, was flying with his magic sandals over the lands on his way back to Greece. On his way, he flew over Ethiopia, but on his way, he spotted a damsel in distress - her name was Andromeda, and she was the daughter of the king of the Ethiopians, who's name was Cepheus. Relevent quote below:

When Perseus her beheld as marble he would deem her, but the breeze moved in her hair, and from her streaming eyes the warm tears fell. Her beauty so amazed his heart, unconscious captive of her charms, that almost his swift wings forgot to wave.—Alighted on the ground, he thus began; “O fairest! whom these chains become not so, but worthy are for links that lovers bind, make known to me your country's name and your's and wherefore bound in chains.”

Long story short, she was being sacrificed to a monster, Perseus slew the monster, married her, her uncle who WAS betrothed to her protested the marriage, causing a battle to break out at the feast, which Perseus stopped by killing everyone other than himself, Andromeda, Cepheus, and the queen (Who's name was Cassiopeia). Fun story!

What that illustrates, though, is that the Romans saw "black" people as...well...people who just happen to have dark skin. They certainly had fanciful tales about what awaited in unknown lands, however, the colour of someone's skin didn't stop one of the most famed Greco-Roman heroes from thinking she was the most beautiful thing in the world. Weirdly enough, it seems like all the artists of that scene ignore the fact that she was Ethiopian :P But ah well. Hope that helped to answer your question :)

I'm not sure if this is precisely what you're asking (and you will get better answers if you specify a timeframe and define what you mean by "black"), but this recent discussion may answer your question.

"Ivory bangle lady" comes to mind. She was mentioned in one of my archaeology lectures and although I can't find the actual research I did manage to find the official press release the University of Reading put out about their finds:

Dr Hella Eckardt, Senior Lecturer at the University of Reading, said: "Multi-cultural Britain is not just a phenomenon of more modern times. Analysis of the 'Ivory Bangle Lady' and others like her, contradicts common popular assumptions about the make up of Roman-British populations as well as the view that African immigrants in Roman Britain were of low status, male and likely to have been slaves."

"To date, we have had to rely on evidence of such foreigners in Roman Britain from inscriptions. However, by analysing the facial features of the Ivory Bangle Lady and measuring her skull compared to reference populations, analysing the chemical signature of the food and drink she consumed, as well as evaluating the evidence from the burial site, we are now able to establish a clear profile of her ancestry and social status.

"It helps paint a picture of a Roman York that was hugely diverse and which included among its population, men, women and children of high status from Romanised North Africa and elsewhere in the Mediterranean."

The ancestry assessment suggests a mixture of 'black' and 'white' ancestral traits, and the isotope signature indicates that she may have come from somewhere slightly warmer than the UK. Taken together with the evidence of an unusual burial rite and grave goods, the evidence all points to a high status incomer to Roman York. It seems likely that she is of North African descent, and may have migrated to York from somewhere warmer, possibly the Mediterranean.

The Ivory Bangle Lady was a high status young woman who was buried in Roman York (Sycamore Terrace). Dated to the second half of the fourth century, her grave contains jet and elephant ivory bracelets, earrings, pendants, beads, a blue glass jug and a glass mirror. The most famous object from this burial is a rectangular openwork mount of bone, possibly from an unrecorded wooden casket, which reads 'Hail, sister, may you live in God', indicating Christian beliefs.

Source.

Pretty interesting but it has to be said one particular instance isn't enough to indicate the average feeling of an entire empire.