A combination of inter-breeding, too much money and power, no-one to tell them to calm down and behave, and a propensity towards serious mental disturbance.
They weren't all crazy. Julius Caesar might have been power mad but was entirely rational about achieving it and Augustus was probably the greatest Emperor the Romans ever had. Only Caligula (definitely) and Nero (probably) could really be dismissed as being mad. They may have all been paranoid, but then there really were a lot of people out to get them!