If you're having problems charging your PlayStation Vita's battery, first of all make sure that the AC adaptor is connected properly.
Connect the USB cable so that the PS logo on the cable is visible beneath the "PSVITA" logo under the touchscreen, and make sure that you're using the AC Adaptor that was supplied with the system.
If you're playing something while the PlayStation Vita system is charging, the charging time will take longer than when the system's not being used.
The PlayStation Vita system can also be charged by connecting it to a USB device such as a PC / Mac. If you're charging from a USB power supply be sure to turn off the PlayStation Vita system before charging it as some devices aren't able to supply enough power to the console.