From a European perspective.
I understand that some would've been random chance. Were they all?
Was there information obtained from natives elsewhere that pointed them towards various islands?
Did they sail in some kind of search pattern looking for islands?
Did they observe weather patterns that don't make sense without land around?
/r/AskAnthropology might be more suited to answering the specific details and 'whys' of your question.
There were several waves of settlement in the Caribbean, starting roughly 7,000 years ago with the Casimiroid people. The latest two pre-Columbian waves, the Arawaks and Caribs, began in the 7th and 13th centuries, respectively. They all seem to have started off at Trinidad and went north from there.
The Caribbean, geographically, is a pretty easy and intuitive area to expand and settle. It's not exactly Polynesia. From South America to Trinidad, you could practically swim there and would be no problem for people with any kind of watercraft. Many of the islands in the Caribbean can be seen from each other and would have been easily known. Expansion would have followed the typical human pattern of taking families to greener pastures or integrating into existing distant societies, which would have been interacting with each other in some way.