A very important question. The answer is that, to the best of my knowledge, there is no official attention paid to racial issues in Jewish discussions. There are some references to people from Africa being the children of Ham or Canaan, but not in the sense that slave owners later used it. In fact, Orthodox Jewish law tends to assume that just about all the national identifications in the Bible are no longer true, that the nations have been sufficiently mixed that we don't think the people who live in x region are actually the usual members of that region (so that Egyptians, for one example, aren't treated by Jewish law as if they are Biblical Egyptians).
It's also true that racial issues in today's Judaism are in some sense less prominent than they might be, because there are many Jews from birth of various skin colors (Jews from Middle Eastern countries like Yemen can be very dark, close to light skinned African Americans; Ethiopian Jews are in fact Africans). I don't say there are no racial issues within Judaism, only that they will be no different for a convert than they are for natural-born dark-skiinned Jews.
More important than any of that technicality, though, is that Jewish law is clear, adamant, and repetitive on the importance of welcoming converts, wherever they come from, and treating them well (in times when we thought we knew lineages, some kinds of converts couldn't marry Jews by birth, but that wasn't a matter of skin color, it was of the actions of the nationality into which those people were born; as I said above, we don't follow that anymore, because we think those lineages have been lost). That's true of all converts; it's not that we should view African-American converts as better or worse, or treat them better or worse, it's that we are required to welcome converts, and love them, and treat them with extra sensitivity for the process they've gone through in coming to Judaism.
Growing up in the nineteen sixties, my mother and I argued about a question which might seem strange from our vantage point today. I asked her: "Which would you prefer: if I came home with a non-Jewish woman or a Jewish woman of color?" Although I would hardly describe my mother as prejudiced, my mother said that she would prefer a non-Jew rather than a person of color. This led to accusations that she was bigoted. In retrospect, I suspect that her attitude had more to do with the difficulties that a bi-racial couple might face in the world at that time.
As a rabbi, I have participated in the conversion of literally dozens of people of color who have joined the Jewish people and embraced the faith of Israel. From the perspective of Judaism, there is no difference between a person of color and a so-called white person. What matters is the depth of commitment of the person who becomes a Jew by choice.
Shortly after I became a rabbi, I was approached by an African-American woman who expressed an interest in becoming Jewish. At the time I was serving in a congregation in the South, so her desire to become Jewish seemed quite extraordinary. For several months we met weekly as she learned about Jewish living, the bible, prayer, history, and she studied Hebrew. One day she came to me with a question. "Was it true that Moses' wife was a Black woman?" I explained that according to the Torah, Moses was married to a Midianite woman named Tzipporah. Howeve,r there is a passage in the book of Numbers (Nu. 12:1) which refers to his wife as an eesha kushit, an Ethiopian woman. (By the way, the sages took this expression to mean that she was very beautiful; this may be the first reference to the contemporary expression that "Black is beautiful!) My student thought about this for just a moment and then face lit up with a smile. She said: "When I convert that is the name I am going to take: Tzipporah." I am glad to tell you that my student when on to become a committed Jewess and an active member of her congregation, even teaching religious school.
The question of race has never even come up as a subject of concern in Judaism, as far as I know. However, there were always questions about converion in general. In the early Middle Ages a man who was known as Ovadiah the Convert wrote to the great sage Maimonides to ask a question. Could he recite the opening words of the Amida, containing the expression "God of our fathers," since, technically, the patriarchs were not his biological ancestors. Maimonides wrote back unequivocally that a convert could recite these words. He writes in a well known teshuvah:
"Anyone who converts [from then] until the end of all generations, and anyone who proclaims the unity of the Name of the Holy One blessed be He, as is prescribed in the Torah – is counted amongst the disciples of Abraham our forefather and they are members of his household, and he has restored all of them to the correct path. Just as he restored the people of his generation through his word of mouth and by his teaching, so too he restored all those who will one day convert with his testament that he left to his children and to his household. …thus, Abraham our forefather is the father of all worthy people that follow in his ways, and he is a father to his students, and they include anyone who converts. Therefore, you should say “Our God and God of our forefathers” for Abraham is your father. And you should say “Who has given to our forefathers” for the Land was given to Abraham."
Maimonides made no distinction based on the race of the convert. To embrace Judaism is to become part of the greater Jewish family.
I am dismayed that in the 21st Century, in a post-Holocaust world, that the question is even being asked, because it questions not the status of Jews By Choice, but the status of those of color. The simple answer is: any individual who fulfills the requirements for conversion is a Jew. The Talmud in Yebamot describes the process, although each rabbinic body has formalized requirements. The essence is: once an individual accepts Judaism, that person is considered as new born, with no religious past, only a Jewish future. But the essential dilemma in your question concerns those of color, or perhaps those who “do not look Jewish”. But I fear that the underlying concern is that even though they might be Jewish can they marry our children as they are obviously of another race. As we have believed through the centuries; Judaism is a religion of choice and not a race. Many of us are of an age to remember “Whites only” facilities or “Christians only need apply”. Many of us defined our Jewish social action posture in the Civil Rights movement. Of course, we are like everyone else, with the same prejudices as well as the same hopes. But as Jews we are instructed to rise above any pettiness. I would hope that once any person joins the Jewish people that we go out of our way to fully integrate him or her into not just our faith community but into our family community as well.