No. You should say either "Who did you go with?" or "With whom did you go?" (but this sounds very formal and old-fashioned).

There are actually a few good grammatical points here but many native English speakers routinely ignore some of them. You only need one 'with' in the sentence. So, getting rid of one of them you would have either 'With who did you go?' or 'Who did you go with?' Because 'with' is a preposition, the pronoun 'who' is the object of a preposition and needs to be in the object case, so it should be 'whom'. After correcting that, you would have either 'With whom did you go?' or 'Whom did you go with?' Personally I think that either of those sentences is grammatically correct but there is a fairly old belief that you should not end English sentences with a preposition. If you follow that rule then of course 'Whom did you go with?" would be incorrect because it ends with the preposition 'with'. That would leave 'With whom did you go?' as the correct grammar. As the person who provided the previous answer points out though 'With whom did you go?' does sound rather formal and I think you would very often hear native English speakers say 'Who did you go with?' even though it's not technically grammatically correct.