The PSP 1001 phat will not charge via USB without a hardware mod. If you're good with small electronics and a soldering iron you can pull the mobo and solder a thin cable from the USB port to the power. Google it.
"Go to settings>system setting>USB charging-turn it on.
Then connect USB cable to psp and to laptop/computer go to settings,go to second options USB connect. All done the green light (near powerbutton) turn to orange colour and your psp start charging"
"If you are trying to charge the PSP with the PSP USB cable, it should work.
Make sure your laptop is charging when you plug in your USB cable to your PSP. Sometimes laptops turn off USB ports when they are running off a battery. If that doesnt work make sure your PSP is off(and both USB ports on computer and PSP are clean and free of debris) and try another USB port on your laptop."