Moving to the caribbean

It seems to me you need to carefully consider your target population: tourists who have the $ and are willing to spend it, and who are hungry.  Stay away from all-inclusive resorts and cruise ships: those tourists have already paid for their food, and they know it.  You want to be near a strip of hotels or condos/apartments where people dont want to be bothered with cooking.  Best example is Aruba. Hardly any all-inclusives. Strip of hotels on NW end of island, compacted, with very expensive food in the resorts.  Right across the street: TONS of restaurants making a killing on US dollars.
For targeting locals, you need to do demographics on the locals, with respect to their incomes.  These statistics may be difficult to obtain or interpret.  I was told St Lucia has a fair number of wealthy people who have moved to the island (now locals).  But for resorts in St Lucia, Sandals (all inclusive) has the lion's share of the market.  I dont see the market there for you.
In my opinion, Aruba is the business model you need.  Go see it! Then look for that type of arrangement elsewhere in the Caribbean.

I have a very similar question to the original poster, I am looking at living the Caribbean but do not have a lot of knowledge of the specific islands and unsure how to decided where to start looking.

We are both English speaking, I am Canadian and she is British. We have kids, and are looking for a clean, safe place with a comfortable lifestyle. I work remotely for clients, so we do not need jobs on the island, just good Internet access for Skype and airline access to the US.

Is there a basic starting guide to cost of living that you could share? House and apartment guides for example?

I wouldn't recommend Belize...