Comprising over 20 island-nations and territories, figuring out where to go in the Caribbean can be incredibly daunting. Unless you have the luxury of time and money to island-hop every few days, I do not suggest visiting them all in one trip.
By and large, the Atlantic hurricane season starts at the beginning of June and lasts until November. Historically, powerful hurricanes have occurred between August and October. To minimize the risk of inclement weather impacting your vacation, I would avoid traveling during these months.
However, the reality is that there are so many airlines that fly to the Caribbean that running individual searches in order to arrive at the cheapest airfare will surely take an enormous amount of time. Hence, I suggest taking the OTA route as they generally scan all airlines, including budget carriers, for the cheapest airfare.
The strategies in finding reasonably-priced lodging will mirror the techniques for airfare. Performing a search using various online hotel search engines is recommended.