you could use an external battery charger for the psp as shown above to charge the battery..
Also you could charge the battery via usb as long as the usb port on the psp is working.
First of all if you have a phat, it is a bit technical and you will need to have some technical skills like soldering.
What it does is basically it allows you to charge the psp battery via usb, without turning on the psp.You could also power up the psp via usb if you perform the usb phat charrge mod.
Have a look, USB Charge by wesley_huddleston1 - PSP Phat Tutorials - Acidmods.com
Now if you have a slim, its way easy and no need to hardmod.
remove battery from psp and connect the psp to ac input
Turn On Your PSP While Holding The R Button Until It Brings Up A Screen With Red Lettering( provided you have cfw installed).
Next Select "Configuration" And Press "X"
Then Select "USB Charge" And Press "X" So It Says "Enabled"
Then Select "Back", And Then Select "Exit"
Your PSP Will Reboot And insert the battery now You Should Now Be Able To Charge It Through A Usb Cable.