There are a number of bus lines along the east coast of Mexico -- ADO being one of the largest, Grupo Senda, Autobuses Americanos, Autobuses Amigos and others. Lots of bus service, and the first class and deluxe buses are comfortable, with reserved seats, videos, snacks, A/C, bathrooms, attendents, etc. but the schedules are complex and unreliable, and connections can be a pain. You can’t just jump on a bus in Nuevo Laredo and get off in Chetumal. It’s about 1400 miles through Mexico to the Belize border, plus the distance from Houston to the border, with lots of stops and connections along the way.
A couple of excellent sources of information on buses in Mexico and elsewhere in the region are http://www.magic-bus.com/
Another thought is that if you can get from Houston to the border via bus, rental car or whatever, and into Mexico you can fly from there to Chetumal. Air service outside of the U.S. is more or less business as usual, and there are inexpensive internal flights in Mexico to Cancun, Merida, Coz or Chetumal that will save you several long days on buses. With all the movement between Mexico and Houston there must be many buses that travel routes to the border.