I recently heard in the brilliant TV show The Night Of, a character saying "he knows not...". As far as I remember my English lessons, the teachers always taught me to say "he doesn't know". Which is correct? Is this an archaic form still used nowadays? Is it slang?
This used to be more common in English (i.e. Elizabethan English like the King James version of the Bible). "Know ye not that....?" instead of "Don't you know...?" So, yes, its an archaic form still used today. It might be a good alternative when you need to avoid contractions but don't want to use the clunky "He does not know." "He doesn't know" is off limits, so you say "He knows not." Especially good for poetry. Consider the famous example of someone picking peddles off a flower saying "She loves me, she loves me not" or "He loves me, he loves me not."
Both are grammatically correct. It's called negative subject-auxiliary inversion.
Although the particular example you gave is an archaic form, it's still acceptable when you're trying to give a statement a more poetic or nostalgic feel. English is much looser on word order than other languages are (or so I'm told, anyway)