Exert normal caution and then some. Belize has a high crime rate, but that should not keep you from what you want to see.
You might want to break up your trip into the Tikal portion and the caye part. I would head off toward Tikal and Guatemala at the beginning to see what I want. Then I would head back to a caye, preferably Caulker to enjoy the ocean.
I think that this web page might help you: http://www.mostlymaya.com/2Belize.html
The worst thing about traveling alone is that there is no one else to notice if you are about to leave some small thing behind. Be extra careful when leaving a hotel room, bus seat or just about any other location behind.
Have a great trip!