Do British Citizens need a visa for Mexico?

No. British citizens do not need a visa to visit Mexico for tourist purposes. You will need to complete an immigration form and have it with you when you arrive and leave Mexico. This form is known as a Forma Migratoria Multiple (FMM). You can complete the form online in advance or when you arrive in Mexico.

Do they ask about criminal records?

The Forma Migratoria Multiple (FMM) does not ask about criminal records.