Being a dual national and carrying the passport of another country is perfectly legal. Under U.S. law, naturalizing as a citizen of a foreign state will not in and of itself cause you to lose your U.S. citizenship. General information about dual nationality is available here.
The naturalization process in Mexico is managed by the Secretaría de Relaciones Exteriores (SRE). There are several methods of naturalizing in Mexico. Generally, Mexican nationality can be acquired by meeting various residency, cultural integration, or marriage requirements. Please contact SRE’s main office for more information on the application procedure.