Make sure that your iPhone 5 is using a genuine Apple Lightning USB cable, and is securely plugged into a USB port on your computer, or ideally a USB power charger that is included with your iPhone 5.
Obviously, make sure all the connections are properly pushed in. If it still does not work, try a different socket, USB port or computer. If it still fails, try another Lightning Cable.
If neither of the above solve it, there may be an issue with your iPhone 5. If this is the case, you're best calling Apple or dropping into an Apple Retail Store.
You can contact Apple here: Contact Us
Hope this helps you out!