If you do a word search for work permits on the site, it will come up with lots of information.
To work legally in Belize you will have to have a work permit, which either your employer will need to get for you, essentially sponsoring you, or you will need to get for yourself. Prices range from $500 to $1500 U.S. per permit, for a year. The process takes about 6 weeks and starts by getting a letter of recommendation from the village council.
Regarding prospects for legal employment, I do not want to be a Debbie Downer, but the economy on Caye Caulker is not too good right now, so most jobs are going to Belizeans, or those who are coming with money to invest in Belize. However, you could get lucky and be in the right place at the right time and have a skill that can not be met with the local work force. It would be best for you to come with enough money to sustain yourself for your stay, and if you are able to find work, then great. It is a small island and everybody knows everyones business if you get my hint.
Ambergris Caye 
AmbergrisCaye.com