Just to make clear, by no mean do I mean to offend anyone by this question, it only occurred to me and one of my friends while listening to Bob Marley. Did black people in Caribic evolve separately from the ones in Africa even if they never came into contact with one another or are the black people that currently live there descendants from the Africans brought there as slaves?
The vast majority of modern-day inhabitants of Jamaica are the descendants of African slaves, the same way that the large majority of African Americans in the United States are descendants of slaves. The Spanish were the first Europeans to take control of the island, as they did in most of the Caribbean. As /u/AncientHistory says, before Europeans there were Taíno people living on most of these islands. They were enslaved by the Spanish. This, combined with disease, led to their essential extinction as an ethnic group by around 1600, clearing the way for the widespread importing of African slaves. The British continued the practice when they seized control of the island in 1655. The coming of sugar agriculture in the eighteenth century exacerbated the need for slaves, as it was particularly labor-intensive, given the technology of the time. It was backbreaking labor, and overwork and disease caused a huge loss of life, necessitating further imports of slaves. Additionally, during the 1655 invasion, many slaves fled to Jamaica's mountainous inland areas, forming autonomous communities of Maroons who harassed the British whenever they could, undermining the slave system.
Slavery was abolished in Jamaica in 1834, as it was in the rest of the British Empire. Jamaica received independence in 1962, as a result of the wave of decolonization after World War II.
The Taíno were related to one or more Central or South American indigenous groups, and so would not have been "Black" in the sense familiar to most Americans today, but rather closer in phenotype to, say, modern-day indigenous people from the Amazon or perhaps Colombia. The Library of Congress has this list of early Spanish sources that might give you a sense of Taíno culture and appearance.
One final note: There were varying degrees of racial mixing between Europeans and Natives in the New World. We call the products of these unions mestizo, and it is undoubtedly true that there are people of Taíno descent still around in these mestizo (edit: in French Canada the term was/is métis) communities. As I understand it, many of them are in Puerto Rico and the DR. Most of the people in the Caribbean today who exhibit a phenotype that a modern American would call "Black" are the descendants of African slaves, or of African immigrants of a more recent vintage.
The earliest inhabitants of Jamaica are believed to be the Taíno, an indigenous Native American people. The population declined after contact with Europeans, and Europeans and African slaves then began to populate the island. The Jamaicans of today are the descendants of these populations. You can read more about the early Jamaicans in The earliest inhabitants : the dynamics of the Jamaican Taíno.
As the other parties have noted, the majority of Jamaicans, as well as other Black people in the Caribbean region, have African ancestry. It is African ancestry that is post Columbus.
Many of these people have other ancestry as well. There is some Native American ancestry and a substantial amount of Western European ancestry. Much of this latter came in via Irish deportees after the Cromwellian conquest of Ireland. Tens of thousands were sent to Barbados and Jamaica as slaves, and they interbred with the Black populations already there.
Most of the African ancestry came from the region of Senegal south to Angola on Africa's West Coast, especially from the corner around Cameroon, where the north/south aligning coast makes a 90 degree turn. So much was from that corner that Mbo people of Cameroon show affinities with some American Blacks; with some guys having the oldest identified to date Y chromosome. These are found now not only in the USA but also in Cameroon.
Some of this ancestry might be from Africa's Eastern shores, but it would be hard to find it, and most of this ancestry would have been from areas within 200 miles of the West Coast of Africa.
Edit: There are a few hints, like Olmec statues that seem to show possible African ancestry, but we do not know if these depictions were of Africans, and if they were, the occasional castaway on New World shores would have not changed the DNA of the much more numerous natives.
http://www.ancient.eu/article/672/
While there are some Black Nationalist theorists who claim that there was a significant population of Black people in the American South before the 1492 era, to my knowledge none of these claims have held up to proof.