If it is being financed you will need to have a letter stating that you have permission from the lending institution to take the car out of the country. You will also need to post a bond using a major credit card. Some Americans bring their car in on a tourist visa and drive up to the border every 6 months to renew their papers. If you don't mind driving to the border every six months then this approach is alright too. You can, of course, purchase a Mexican car and get it legally registered no matter what type of visa you have.
Car rentals are very expensive in Mexico.
The major US car rental companies have offices in Mexico. Be sure that they inform you of which papers you need to present if you are stopped by a police officer. Make sure that all the rental car permits are current. This includes services and restaurants. Because of the low wages in Mexico, tipping is an important part of the worker's income.