iPhone 5 only charges when turned on

Hi, I have a strange one here with a customers phone. The phone came to me after someone else had changed the battery on it. The phone wouldn't come on and wouldn't charge and the other person refused to resolve the problem. I changed the battery and dock connector, charged the phone up to 100% as it had some juice in the battery anyway and gave the phone back. The phone was returned to me the following day with the customer saying it won't come back on. I tried it with my charger and had no luck, I then removed the battery and tried it in another phone and it charged fine. I returned the battery to the phone and it would charge just fine whilst turned on. I left it plugged into the charger and it stayed on fine. I turned the phone off and left it unplugged only to find it was dead again the next morning. It would also not charge either, so repeated the battery charging again and it's now got the battery back in and is charging again, so two issues here, one.... It goes flat when not turned on and two once the battery is dead the phone won't charge it unless its turned on ?.