I agree that it is most likely the battery, but it could also be the charger, or a combination of both the battery and charger.. Quick chargers also tend to make the batteries heat up more. Try a slower one.
In this case, the phone runs with either the AC adapter or the battery, or both. Whether it runs or not, though, is not the question here. It overheats while charging. Which means it could be either the AC Adapter not charging the phone correctly, or the battery not taking a charge correctly. It's more likely to be the battery than the charger, though.