It does not trigger. It goes by what's already logged for you in any previous games playing Suleiman. I had played him once before and had captured 7 barbarian ships. I waited until one popped and converted and did not get the achievement. When I checked my log files it had incremented barbarians captured to 8. It's a total number of play throughs so you do not have to keep them alive. I'm not sure what would happen if you captured one then promoted it but I suspect since it increments the counter on capture you should be fine.