There is a distinction between classical slavery (enslaved in warfare or because of debt) and trans-Atlantic slavery (commercial slave trading of specific ethnic groups). East Asian slavery is similar to the former; the Muslim-controlled slave trade across the Indian Ocean and within the Ottoman Empire is similar to the latter.
The Slavery/Forced Labour civic surely represents classical slavery - we don't use it for America's spawn, do we? If you wanted to beef it up, then you could bestow the Aztec UP more widely.
Trans-Atlantic and Indian Ocean slavery is currently unrepresented. Here is a scheme that would be relatively simple to implement:
1. Cotton, coffee and sugar resources require a new Slave Plantation improvement [identical graphics to current Plantation]
2. The Slave Plantation can only be built by a new Slave Worker unit [
graphics here]
3. Only appropriate civs will be able to build Slave Workers: perhaps America, Arabia, England, France, Mali, Portugal, Spain)
4. Create African Labour resources in west Africa, and perhaps the Great Lakes region. The (very cheap) Slave Worker unit can only be built if you have this resource.
5. African Labour resources require a new Slave Market improvement [camp graphics, as merijn_v1 said]
6: Most of these civs already have appropriate areas in their settler maps, but double-check everyone except the Americans does.
Problem: Relies on others selling African Labour to Americans. Will an American AI trade for it?
Some nice extras:
7. Slave Workers should always be killed by plague, and should preferably die when creating the Slave Plantation improvement
8. Slave Plantations generate local

[using the same mechanism that generates

from jungles]
9. With an appropriate civic/technology/event, then you can build Engineers, which obsolete Workers and Slave Workers
Problem: Needs to incorporate a liberation event
Problem: The removal of Emancipation means that there is no longer an obvious civic for Engineers
A more historical way to generate Slave Workers: Make them poppable from huts, and have large numbers of huts appear in west Africa/Great Lakes in the appropriate time period.
Problem: how to get AI to repeatedly explore these areas.
Advantage: huts elsewhere represent classical slavery and Spanish enslavement of native Americans
This slave mod might also be useful for some other scheme:
http://forums.civfanatics.com/downloads.php?do=file&id=13322