Claudio makes an important decision in Act III. He decides that there is concrete proof of Hero's infidelity. He does this based upon events that happen offstage and only reported to the audience in Act III, Scene iii by Borachio, Don John's henchman.
The only scene in which Claudio actually appears in Act III is Scene ii, in which he first (along with Don Pedro and Leonato) teases Benedick about cleaning himself up and shaving, presumably for Beatrice. They continue the good natured, friendly banter that they have shared up to this point in the play. By the way, this joking and bantering about will take an ugly turn in Act V. But in Act III, all still appears to be fun and games amongst the three comrades.
Then Don John enters. He is Don Pedro's half brother and has sworn to ruin things for Claudio. His main motivation for his dastardly acts? He calls himself "a plain dealing villain." So basically, he just likes to stir up trouble. Already, Don John has tried to convince Claudio that Don Pedro (seen dancing with Hero, his face hidden by a mask in Act II) was making a play for Hero behind Claudio's back. And though Claudio gullibly took the bait then, but was proven wrong, he shows himself to be still susceptible to Don John's accusations here in Act III. Don John basically says that he has proof of Hero's infidelity, and will take Don Pedro and Claudio to stand beneath her window so that they can see for themselves.
Though sometimes staged as a dumb show in performance, the scene in which Claudio and Don Pedro watch the pretend Hero (played unwittingly by Hero's serving woman Margaret) engage in a tryst with Borachio is not included in the play. So, the audience is left to wonder why Claudio would be so foolish as to believe Don John, who has already proved himself suspect with his first accusations of Hero. Why do you think he believes Don John? Shakespeare doesn't give us any definite motivation, and seems to have purposely left this very important event -- the witnessing of "Hero's" infidelity out of the play. A couple of things about Claudio, however, can be said. He appears to be easily made jealous, and he's pretty gullible to boot.