Every person with Mexican nationality is legally required to declare himself/herself as a Mexican national when entering and leaving the country (Art. 12 Ley de Nacionalidad). Mexicans who hold other nationalities and that enter or leave the country declaring themselves as foreigners can be fined with anything between around 21000 and 35000 pesos (Art. 33-I Ley de Nacionalidad), so OP you should never identify yourself as a US citizen when entering or leaving Mexico.
Mexicans don't need a passport to enter Mexico. A passport can be used, as well as a matrícula consular, voter card, certified copy of birth certificate and other documents that prove Mexican nationality (Art. 36 Ley de Migración). Foreigners are explicitly required to enter with a valid passport (Art. 37 Ley de Migración).
Mostly but not 100% correct: Mexican law allows non-Mexican nationals of multiple other countries to enter with other than passports. For some only a birth certificate and photo ID is enough.
My children have entered and left Mexico using either, and even both, of their American and Mexican passports - never been a problem. One time, back when my wife refused to believe me on this issue, a Mexican immigration official agreed with me
over my wife that they should have used their Mexican passport to enter Mexico but she (the official) did not make them change their entry documents.
I advise against mixed use of travel documents, though, because it can cause quite the administrative headache when one is trying to leave and the airline/la migra is trying to collect the FMM card and it doesn't exist....
Also, using the correct document to enter
can have an impact on a traveler's treatment and rights in legal situations because, in theory if not practice, the citizenship claimed on the documents used to enter a country determines the person's status.