There are a couple ways of going about this. Flights are a joke of a price and considering the close proximity to one another, a waste of money as well.
From the airport there is a direct bus that leaves daily at 2215 with the bus company ADO. This goes straight to the bus station in Belize City and costs 498 pesos, arriving some time the following morning. From the bus station, taxis are 7 Belizean dollars to anywhere in the city, these are basically cars with a green number plate, they aren't exactly labelled as taxis.
Another option if the night bus is fully booked (this is what happened with me) is to take a night bus from Cancun to the border town of Chetumal. There are a few buses run by the ADO bus company and I took the one at 2238 from the downtown terminal in Cancun, it takes six hours to Chetumal. This bus costs 302 pesos. Of course, this bus ride can be done just as easily during the day time (and I'm sure there are other companies), I am merely describing how I did it at night so as to save a nights accommodation, very important to most backpackers.
Once at the Chetumal station, you can pay 200 pesos for a shuttle bus/ mini van to take you onward to Belize City. This is with the company San Juan, and they have hourly rides from 7am onwards, I'm sure there are other companies operating out of the station. The minivan stops at the Mexican border for exit stamps at Subteniente Lopez, additional tourist tax may be required here from certain nationalities. I read somewhere that it is a hoax and a way of ripping off people but they seemed pretty insistent and handed out stamped receipts, possibly not legit but not too bothered and not much you can really do is there? The tourist tax is 294 pesos.
Entry to Belize through Santa Elena is plain sailing and as they speak English you have a welcome greeting, provided English is your native language, stamp in your passport where you request it and then it's back on the van to Belize City. Nothing much to see along the way and by 1030 you'll be dropped off at the car park opposite the Caye Caulker Water Taxi jetty which is perfect for most people who are literally heading straight to Caye Caulker.
NB: Whilst driving south in the minivan to Belize City, we passed a couple buses heading north to Chetumal. These were local Belizean buses which no doubt cross the border and go to the town of Chetumal. I don't know if they go to the station where the ADO buses arrive and depart from but someone at the station would be able to tell you where to get one of these buses which, by basic reasoning, must then return to Belize City for what I can only assume of a cheaper price than 200 pesos. If your budget is that stretched, this could be another option.