First off, if this is in the wrong place, please feel free to move it to the right place. I have an iPad 3 64g 4g. When I plug it into any computer, I get "not charging". When I plug it into the wall, the brick in the wall and the usb plug in the brick heat up to concerning temperatures. First, why won't it charge through the computer? Second, why is it heating up so bad when connected to the wall?
Regarding the first point, USB was never designed to provide enough power to charge devices such as this. It will charge slowly with the screen off but not with it on. In fact I think even my original iPad was this way. I won't comment on the second part because it's totally subjective. What might be concerning to you could be normal to me.
It's a lot like the light in your refrigerator. It's not on all the time, only when you open the door, but you can't see when it's off. The "NOT CHARGING" on the iPad, when hooked up to the computer, is only because the computer can't charge and have the screen going at the same time. When the screen is off, it'll be charging, but slower than the standard plug that came with it. You'll just never be able to see "CHARGING" when hooked to the computer's USB port, a la the refrigerator.
When you are charging it and it is getting pretty warm, are you using it at the same time? When I had my AT&T iPad, it heated up a lot when charging and using it at the same time. I exchanged it for the Verizon one for the better data speed, but the new one doesn't do it.
iPad will charge very slowly from USB Plugged into computer. It is normal for charger to get warm while charging. Mine always has.