It should be "teach". "Let alone" is functioning as a conjunction here, with a meaning similar to but more emphatic than "or".

Yeah, the correct is "teaches", just because you're still telling about the pronoun "he" and you don't have any auxiliar verb to put the verb "teaches" on your basic form: "teach". It's the same if you say: "He doesn't speak English let alone he teaches it." or "He doesn't speak English let alone does teach it." On this last case, you can put the pronoun or not, but is better put the pronoun in.