To unlock the Duel-icious trophy in Borderlands: Game of the Year Edition, you will need to be playing in co-op. Both online and split-screen will work.
Walk up to your co-op partner and melee them to propose a duel, and have them melee you back to accept. Kill them and you will unlock the Duel-icious trophy. If you're boosting this, ask your friend not to attack you back and you will also unlock the trophy below: