iPhone won't turn on while charging, get's hot!

This morning, I notice that my Iphone 4 was quiet hot and din't charge fully from its one-over-two-days overnight charge (only 73%, which felt really weird !).

Today, it seemed to lose charge a bit quicker than usually but nothing dramatic; and fell short of battery tonight.

I have been trying to turn it on for an hour now (half an hour charging on sector, half an hour on my macbook) and I can't get it to turn on. When plugged in, I do have the charging screen with the red battery and the white lightning bolt.

I must add that charging makes it really hot (I would say 30-something °C) on the upper,right-hand corner.

I tried hard-reboot (both buttons for 15s) and recovery-mode and both failed.

Now, I understand that if I were a dead battery, it would only be a couples a dollars.

How can I troubleshoot my problem and, hopping it is the battery that is faulty, be sure of it?