I went travelling around Italty and I found Naples
terrifying.
As soon as I got off the train someone through a bottle at me and groups of men were shouting at me to come over to them, my tourist nature given away by my large backpack.
I eventually found my way to the circumsuvienna (sp?) metro line and got on, befriending an Italian teenager for my own comfort. Within 3 stops there were people dealing drugs in my carriage and eventually a large gang got on. My Italian chum whispered at me not to look at them, and then eventually told me they were planning to mug me
He whispered to run off the train at my stop as the doors were closing and not stop running until I got home. He got off a few stops before me and true enough several members of the gang were staring at me and one of them had a chain!
I ran off the train and sprinted all the way to the hostel on which point I collapsed to the group and pretty much kissed the floor for thanks of safety
Bear in mind, this was me getting into central Naples at night by myself - very vulnerable.
Otherwise, the city is a true dump, I haven't seen anything like it in the UK. Graffiti everywhere, rubbish littering the streets... but there are some interesting things to do. You can take a tour of the underground Roman aquaduct that runs under the city, which was really fun.
You will be much better off staying somewhere like Sorrento and taking a day trip or two to Naples. The amalfi coast is really special.
Oh and don't get the night train from Naples to Sicily. I bought a ticket only to find someone people who had their bags stolen from them at that train by being gassed unconcious and their bags removed from them by knife

... if that hasn't put you off I don't know what will
On the plus side, the outskirts of Naples was the only time in mainland italy I found myself in a culture that didn't pander to tourists, I stumbled accross this truly local market with stores being run by kids and grandparents. That was actually one of my hilights of my trip. But generally, stay outside naples and spend more time exploring the Amalfi coast / Capri etc.