When you plug in the charger without a battery connected, the iPhone starts normally, maybe the problem is in the flex charger lightning.
I finish to repair a IPhone 5 that would not start, I try doing the "hard reset" the final problem was the power button flex and battery. If the power button does not work you do not get to do the hard reset, but you can do it, holding down the home button, and connect the charger "USB connected to the computer" in this way enters restoration ele mode.
If not, the problem is in the charger lightning flex, try a new, and try to power the iPhone, if it does not work, try changing every flex? Perhaps one is giving a short circuit on the motherboard.
If not, then the problem is on the motherboard, you'll have to find some qualms technician for motherboard.