The best time to go sightseeing in Naples and enjoying its beauty to the fullest is probably spring or late summer - May, June, or September. The weather should be warm and pleasant, even if a couple rainy days can happen.
June and August are the best months for going to the beach and swimming, but they can be very hot. Plus, the city might feel a bit empty, since many Neapolitans go abroad on their summer holidays.
Christmas time in Naples be a surprising experience - the city gets crowded with street markets and nativity-themed stands.