Apparently there were some coins found from Sinope that were found to be vandalized. However, there is no doubt that Diogenes wanted to perpetuate the story that he actually damaged the coins of his city of birth to protest authority and custom resulting in a trial and exile.
True or not, defacing the currency is an apt metaphor for the kind of life advocated by Diogenes. 
He wanted to break through the hypocrisy of everyday thoughtless morality, display convention and habit as shallow, and upset the powerful and self-satisfied citizens of Athens. I am reminded of the yippies of the 1960s and 1970s who lived simply, maybe in primitive communes like the hippies but sought to parody and poke at middle class, humdrum lives and pieties. They would sponsor slum tours, only not of slums, but of middle class suburban neighborhoods showing the boring habits of people devoted to their lawns and Buicks. Defacing the currency. Unlike the yippie poseurs, Diogenes lived a life of sparse, natural existence and showed that he believed enough of his own philosophy to promote it be demonstration. Here is a picture, one of many similar paintings, by Jean-Leon Gerome (from wikimedia commons) of Diogenes living in a jar, lighting the lantern he uses to search for one good man. He is admired by a troop of dogs, who gave him the name of his philosophy, four fine looking Cynics.
Two Strikes 