To start a duel with another player simply join a co-op game and then walk up to another player and melee them to propose a duel. They will need to melee you back to accept the duel.
This is an extremely easy achievement to boost with a friend as it only takes a few seconds to complete a duel. If you are boosting with a friend, ask them not to attack you back and you can get the Duelinator achievement at the same time.
This can also be done in splitscreen mode if you have access to a second controller. If you don't have a friend to boost this with or a second controller, try to learn the strengths and weaknesses of each class as you play co-op and use that to your advantage. You can also try to start a duel with another player that is a few levels below you and that will give you a strong advantage.