Bob,
Most car rental companies in Mexico will not allow you to drive over the border and into Belize. I know that only one does. I think that is EuRopcar. And you would be required to get insurance for your trip into Belize. So be sure to check this before you rent. It may be easier for you to rent in Mexico, then take a bus to Chetumal, cross the border into Belize, then rent a car in Belize separately, as David mentioned. Note that there is a comfortable ADO bus that travels from Cancun all the way to Belize City. I don’t think that one way bus ticket would cost more than about $30 US.
The best car rentals in Belize can be found at the international airport or in Belize City. So if you take the ADO bus you can then rent a car in Belize City, where the ADO bus route ends.
As David mentions, you can also fly on Tropic Air from Cancun to Belize International airport. But that flight is quite expensive, when compared to the bus. The bus takes about 8 hours though. The Tropic Air flight from Cancun to Belize City International is currently priced at $200-$265 one way, depending on the day of the week. That seems very steep to me, but it’s because you have to fly between two international airports. So the taxes are steep on that ticket.