They were two mobsters of the Red Hood Gang who hired the man who would become The Joker for the robbery of a card factory located near Ace Chemicals to pose as Red Hood and inform them about Ace's structure to sneak into the card factory, but it went all wrong and ended up setting a chain of events which culminate with the man's transformation into The Joker.
Vinnie was voiced by JP Karliak, who also voiced the Music Meister and the Nome King while Joe was voiced by Andrew Kishino, who also voices Janja.
Years before The Joker's failed attempt to drive James Gordon into madness, the man who would become The Joker comes in contact with Vinnie and Joe. An expecting father, he was looking for a job to support Jeannie, his pregnant wife, and as a former employee of Ace Chemicals, Vinnie and Joe hired him so he could instruct them how to infiltrate Ace and thus be able to sneak into the card factory located nearly. They come to an agreement and then reveal to him that he will pose as Red Hood: their gang actually had no leader; instead, they just hired an accomplice to pose as the mastermind and then performed their operations. The man doubted, but Vinnie & Joe ultimately convince him by asking him if he wants to raise his son in poverty.
The day of the heist, lying to Jeannie by saying that he was going to a nightclub to audition as comedian that smae night, the man meets them at the same bar where they first talked about the heist so they can do the final preparatives. Unfortunately, an inspector and a policeman arrive to the scene and inform the man that Jeannie accidentally electrocuted herself to death in a household accident, thus killing too his expecting child. Deprimed, the man tells Vinnie and Joe the news, to which they express their grief over the matter and decide to leave him alone until night, still planning to perform the heist. The man tries to desist, but they warn him that it's too late to change his mind and indirectly assure him that if he desists, they will get him killed, fearing that he could report their plan to the authorities and promising him that with the stolen money, he would be able give Jeannie a luxurious funeral.
That night, Vinnie puts the red hood over the man's head and they infiltrate into Ace Chemicals. Unfortunately, a policeman appears on the scene and yells them what they are doing there, as the security appeared to have changed since their Red Hood's resignation. Realizing that their plan had failed, Vinnie shoots at the policeman, with no success. They start to run away from the policeman, who calls his fellow officers. While running, Red Hood complains not be able to see good with the mask on, leading Vinnie to promise him to be killed once safe. However, policemen shoot at the trio, shooting Vinnie on the head and Joe on one of his legs. With Vinnie dead and Red Hood unable to see good, Joe tries to give the blame to Red Hood while taking his gun to shoot the policement. Despite this, Red Hood escapes from the police and Joe gets killed too by the authorities. Batman later appears to stop Red Hood, but he escapes from him by jumping into a chemical vat, transforming into The Joker.
Back in the present day, The Joker remembers the instance in which he unsuccesfully helped Vinnie and Joe to steal the card factory. Later on, however, when Batman arrives to defeat him once more after having tortured Gordon, The Joker muses about his past, affirming to the Dark Knight that he sometimes remembers his past different, even saying that if he can have a past, he wants one with multiple choices, thus possibly leaving in question the existence of both Vinnie and Joe. However, if such story is true, there's one certain thing about Vinnie and Joe: they are indirectly what caused The Joker to become in what he is today.
The names "Vinnie" and "Joe" are never mentioned neither in the one-shot nor in the film adaptation. Instead, these were revealed via Alan Moore's script, as the names of other characters of the story.
Furthermore, in the film's screenplay and credits, the characters are referred as "Mitch" and "Murray", but "Vinnie" and "Joe" are considered the canon names as the comic book is the original source.
Despite what many people believe, Vinnie and Joe are not the secondary antagonists; Huey, Dewey and Louie amongst the other circus freaks are. However, Vinnie and Joe serve as the posthumous Big Bads (assuming that The Joker's story is real as heavily implied), because their actions led the man who became The Joker into becoming what he is today by forcing him into be their accomplice, thus indirectly causing his encounter with Batman and subsequent chemical vat.
In the video game Batman: Arkham City, The Joker mentions to Batman on his cell that he remembers the events of The Killing Joe and recalls that Vinnie and Joe worked for Carmine Falcone.