My husband's great grandfather was Jamaican and settled in Liverpool in 1860. We know this from census returns and family photographs (and the fact that we have a huge part of our family who are black). I have found it impossible to get any further back because I have no idea which parish in Jamaica the family was from or who was the slave owner. I can only assume that Lawrence was the name of the person who 'owned' our Jamaican ancestors. You need this information to be able to track black Jamaican families.
There is a very good website you can try jamaicanfamilysearch.com Its full of useful information and lists of slaves, manumissions, slave owners, newspaper reports etc. but without a name and place it is very difficult to track black Jamaicans back to Jamaica.
The BBC website, if I remember correctly, also has some information on how to trace ancestors from the Caribbean.
Good luck.
Ruby