You know, sometimes I feel like Capt’n Jack Sparrow, walking around a pristine white beach with a bottle of Rhum in one hand and a biiiiig smiley on the face. So what about going to the Caribbean and well… You know… Try to do it the same way? 
How to get there
From Germany there are some direct flights from Frankfurt operated by Condor. All others seem to require at least a stop-over, either in London (British Airways, Virgin Atlantic), New York (JetBlue, among others), Charlotte (US Airways) or Miami.
Once there, the best connections over most islands seem to be from LIAT. Another way would be by taking ferries, which connect some of the islands.
[mappress mapid="4"]
Itinerary?
An idea would be to fly to Barbados (relatively cheap British Airways flight), explore the island, and then fly with LIAT to either St. VIncent and the Grenadines, Grenada or some other Island over there.