Naples in the Campania region of southern Italy exhibits a typical Mediterranean climate. The weather turns quite hot in Summer; cool and damp in the winter. It can be foggy along the coast in winter, although there will also be bright, sunny days which will be tolerable due to the usually mild evening temperatures.
If you're interested in air quality in Naples, you can find some good data over at Plume Labs.
Avoid July and August if you can--Italians do. Notice not only the highs on the temperature chart, but also notice that the night temperatures in summer don't cool off all that much. In a hotel you run the risk of sweltering without air conditioning. You're better off renting a house or apartment that you can get an airflow through.
The best time to visit is April-June and September-October, when temperatures are the most pleasant.
Lots of visitors will choose December for the Christmas festivities. Bring a warm overcoat to ward off the occasional icy wind during this period. The Christmas festivities generally start on December 8th, the Festa dell'Immacolata Concezione (the feast of the Immaculate Conception), a time when the Christmas cribs are brought out and built. The season ends at Epiphany, January 6.
Via San Gregorio Armeno is where people go to populate their Christmas crib or presepe. During the Christmas period there will be many people in the streets; you might find it better to visit during November, as business is picking up but hasn't peaked. Most of the stores are open all year around.
If you like fireworks, plan your vacation around New Year's eve; Naples has the best fireworks in Italy.
Easter is also a busy time in Naples. The religious Naples holds its big festivals in September,the Festa di Piedigrotta on the 7th and San Gennaro on the 19th.
The Amalfi towns of Positano, Ravello, Sorrento and Amalfi are less than 50 miles from Naples in the region of Campania. While the tourist season starts around Easter, May and June prices can be lower, and May is likely to be your best bet for a colorful, uncrowded visit to the Amalfi Coast.
Music lovers will be Attracted to Ravello, whose summer concerts were so popular they've spilled over to the rest of the year. Check the shedule at the Ravello Concert Society.
Despite the fine summer weather, the stunning coastline is clogged with tourists from all over in July and August. Try April and May or September and October to slow travel the coast. The ferries and hydrofoils operate from April through most of October.
Amalfi coast temperatures are comfortable even in Winter, although it will be wetter.
A great way to enjoy the Amalfi Coast is to rent a vacation apartment (preferably with a view of the sea). A hot spot on the coast is Positano, and VRBO lists over 450 Positano Vacation Rentals, many with views of the sea. Outside the season you'll find many deals on vacation rentals on the Amalfi Coast.
An apartment you say! But I'm tired of cooking! Well, yes, there is that. But Compania is a prime area for food. There are many restaurants willing to serve it to you. But there's another idea. A new idea. Why not go to the house of a "home chef" and have dinner? Check out BonAppetour Naples/Amalfi.
The map of available hotels and apartments is centered upon Naples, but you can zoom out to see the Amalfi Coast.