Most Final Fantasy characters are racially ambigious characters with both Caucasian and Asian features. They are designed to look mixed race to appeal for both Asians and the west.
Final Fantasy do have Black characters through out it's history. I will only list down the playable ones.
Barret from Final Fantasy 7 is a playable character. He is also one of the favorites. You can customize him as the leader instead of Cloud if you like.
At Final Fantasy 8, There is Kiros. He is playable once you got to Laguna's arc because he is Laguna's partner.
Final Fantasy 12 have Fran. She is a Black woman with rabbit ears.
She is an archer and she can also cast spells. She is a playable character who is relevant to the story.
Sazh is one of the major characters in Final Fantasy 13.He is also the oldest playable character since most playable characters are teenagers to early 20's. if you love gun slinging characters you will love him.