Much Ado About Nothing is the comedic version of Romeo and Juliet, except with two sets of lovers (Hero and Claudio; Beatrice and Benedick). Both plays have a Prince (Escalus and Pedro). Both plays have a villain (Tybalt and John). Whereas Romeo and Juliet's villain is killed in a fight in Act III, thus dooming Romeo, John's plan to have Benedick kill Claudio is ironically spoiled by a buffoon, Dogberry. It is this turning point in Act III which moves Much Ado from tragedy to comedy.
John's villainy is rather unmotivated. He is kind of like Iago-lite. We don't know why he wants to take revenge on Pedro. John is a Bastard, a stock character, an archetypal villain who is jealous of those who are legitimate, like Pedro. Bastards, as you know, are dispossessed sons. They burn with resentment. John can't have what he wants (whatever that is), so he lashes out to hurt anyone who is happy. His deeds are more for effect--he wants to provoke action in others. Usually, a Bastard proudly announces his rebellious dealings; all except John. He is such a mopey sourpuss about his role as a Bastard that it comes across as funny.
So, John attempts to destroy Pedro's reputation: misery loves company. But, unlike Iago, John goes about his revenge in a very indrect way. So says eNotes: "In addition to his attempt to destroy his brother's reputation, he also attacks Hero's honor and happiness." This villainous attack on reputation brings together both sets of lovers in the end. Claudio is reconciled with Hero, and Benedick moves from being a sexist pig to a gentleman--nay, a chivalrous defender of the softer sex.
Everyone is fooled by John's ruse: Pedro, Benedick, and Claudio especially. It is ironic and thus comedic that Dogberry, a bumbling constable, should uncover this plan and not the noble characters. Comedy is thus achieved by subverting the natural order: a low class character unwittingly upstages and upper class characters.
Dogberry saves the play from tragedy. Claudio does not kill Claudio. Benedick comes across as noble for defending Hero's honor, and Beatrice marries him for it.