Look at it this way: you were convicted of a DUI six years ago (by the time you apply). Is this a black mark? Yes. Will schools toss your application because of it? No, not if you have a very strong application. If you have a borderline application and they're on the fence about you, a DUI conviction would probably be a reason to say "nay".
Will it come up in an interview? Probably. But if you get to the interview, they're presumably interested in you and it's how you state your case. Have a good mature story to tell.
Drinking infractions are one thing. DUIs are a whole different kettle of fish. If a medical school thinks that you'd be prone to have one again, I think they'd be insane to accept you. So what can you do to convince them it will never happen again? Counselling, community service, volunteering at MADD or somesuch, etc. You should have a better story to tell than "Whew! That DUI was bad news. That'll never happen again." Because this is what all folks who get DUIs say and many of them do a repeat performance.
Work hard to make your application more competitive than the next person and learn from your mistake and you will do fine. A DUI may keep you out of some med schools, but not all and all you need is one.