My iPhone 5 is completely dead.

While using a third party cable, the message about it not being verified by Apple and may not working reliably came up and after that it started to drop its charge, while still charging! After going down and down, it proceeded to die completely. After being put onto charge again, it finally opened (at 76% battery) after I used the power button at the top, before dying again. Now, it shows that it is charging while plugged into the charger, but won't turn on at all! As soon as you take it off charge it flashes the little usb icon. It won't connect to iTunes and it won't reboot when I hold down the home and power buttons. Is this issue likely to be with the third party cable, or with the battery? Has anybody has a similar experience.