How does a convicted felon receive a passport for travel to and from mexico?

They fill out the application, send it in with a photo and the proper fees, there are no restrictions on getting the passport, The restrictions comes from having the criminal record when and if stopped to have ID checked.
And in other nations where you require a visa to enter.

Does one need a passport to go to Mexico now?

I called ( planning a trip) and I was told yes.