To the best of my knowledge, Duffman has only made two appearances on the Simpsons, but both of them were classics.

Appearance the first:  Moe informs the barflies at the tavern that they are the cause of most of the accidents in Springfield.  The guys begin to celebrate the perceived good news, but their celebration is quickly haulted when Moe informs them that they now have to practice the designated driver program.  To determine who will be the DD, each bar fly must take a pickled egg from a jar.  Whoever has the black egg has to be the DD.  Much to his chagrin, the unluckly holder of the black egg is Barney Gumble.  Barney is seen eating peanuts and sweating nervously, when who should appear?  The one and only Duffman.  Duffman announces that he is at Moe's to party with Barney Gumble.  Barney informs Duffman that he is the DD to which Duffman replies that he supports the DD program and asks the other patrons at Moe's if they are ready to party.  Duffman parties with the bar flies long into the night, including Homer Simpson, a.k.a. Peter Pantsless.  In the end Barney does his duty and drives the barflies home.  Duffman's story ends here, but Homer's is just beginning.

Appearance the second:  The Simpsons awake early in the morning to go to the Duff Beer festival, where Moe Syzlack is competing in a bar tending competition hosted by Duffman.  Duffman announces the names of three competitors:  some mick from Chicago, and the beautiful Titania, and Moe.  The first competition is trick pouring, which is won hands-down by Titania.  The second is the trivia competion, where Duffman says that Duff is made from "fresh mountain what?"  Duffman determines that Titania's answer of "goats" is close enough, as the fair maiden sweeps the first two round of competition.  The final round is the drunk toss, which Duffman announces counts for 98% of the score.  The mick first heaves his drunk a great distance.  Titania refuses to toss her drunk, saying that she shouldn't have to toss the drunk, because she slept with Duffman.  When Moe has to throw his drunk, a.k.a. Barney, he grabs a stick with a nail through it and chases Barney the distance to win by a tounge's length.  Duffman announces that the winner of the competion is Moe, who gets to have his face printed on the Duff calenders.  When the calanders finally come in, Moe discovers that Duff had placed several stickers on the picture in order to cover his ugly mug.  Taking the advice of his friend, Carl Carlson, Moe gets plastic surgery on his face on goes on his revenge tour, spiting everyone who ever hurt him for being ugly.  On this tour he sneaks up on Duffman, who is minding is own business, and slaps a sticker on his face, rendering him breathless.