If you can boot up the phone and still can see the screen even though its broken try to get a micro-usb to usb female cable and plug a mouse to it, ive done that and i could use the phone with the mouse and send pictures through some program that let you transfer pictures and data via wireless
Have you tried calibrating the battery? Or changing the battery? That could be a malware or some type of issue like that, you can try replacing the battery, trying a less powerful charger and completely reboot and restore the phone.