How do I get a PS4 controller to charge?

If by meaning you've tried charging it using the cable and it won't charge or hold a charge then the internal battery has gone bad.

Consult Sony PlayStation support, I don't know if the battery can be replaced or not.
I assume the controller is serviceable since there are screws on the underside but i never took one apart.

What do you mean how?

You just plug the cable into the USB port in your console and the other end into your controller.

Buy a new one