Could be a number of things.
Try turning the phone off then charging it. If that works, then the battery and USB port are fine and the issue's probably software. If it doesn't work then you either need a new battery or a repair. If the battery is removeable, try borrowing a battery to see if that works. If it doesn't, the phone will need to go for repair.
If the phone does charge when turned off, the first thing to try is simply checking what's using the battery - tapping Battery in Settings should give you the information. If there's one particular app that's using the all the juice, try disabling it, see if that fixes the issue.
If that doesn't help, try disabling everything you installed or updated around the time the problem started.
If that doesn't work either try disabling all your apps then re-enabling one by one until you find what causes the problem or back everything up and try a factory reset.
