I've never been on a live aboard, but I'm sure others will chime in.
You might want to take a look at the resorts out on the atoll as they offer some fantastic diving:
Turneffe Island Resort,
Turneffe Flats or
Blackbird Caye Resort. These are probably beyond your budget but keep in mind their prices include food, lodging and diving so they aren't as expensive as they might at first appear. I see TIR is currently running a special, which makes it even more affordable. We've been there twice for a total of three weeks and it's our favorite resort on earth.
The majority of tourists and divers head to San Pedro on Ambergris Caye. It's got more of everything like dive ops, accommodations, restaurants, night life and, of course, people. If you want something a little smaller take a look further south at Placencia. It has more of a fishing village vibe. The dives are a little further out, but in my five or six weeks there I don't remember anyone ever getting seasick. I love Placencia because even though it feels like an island it's actually connected to the mainland and there are all sorts of jungle related things and sightseeing opportunities right up the road.
If you can narrow down your focus a bit--Belize is a small country but still a big place--you can get lots of specific recommendations.
A good way to get to know the country in advance is get the travel guide
Belize Insight Guide. It's a general travel guide so it doesn't have a lot of dive specific information but it's chock full of color photos, maps, and district by district information.