IPhone 4 battery or circuit problems?

I noticed a few weeks ago that my iPhone battery was draining extremely fast and then today it got Really Really hot and the battery completely died. I didn't even use my phone today I just turned it on. I think it might be a short circuit or the battery. I can still plug in my phone and it will turn on but the top right side of the phone gets extremely hot and then it spreads to the rest of the phone and my battery says it's "charging" but doesn't actually charge and I've seen it actually loose battery life while charging. It will not sustain life after it's unplugged. I do not have anything running on my phone as well as no calendar or any other updates/software running and I have my Wi-Fi and data turned off and it hasn't changed. It all started when the lock button got stuck, then the vibrating motor broke and now my battery doesn't work. If I do need to get my battery or my circuits changed how much would that cost me from Apple when I'm not covered under warranty?

You have what is probably a bad charger circuit or charger port with pins shorting or both. It is the phone, not the battery unless you leave it on until the battery fries too. Heat in a phone can be a shorted battery, but in the way you describe it, it isn't.

I have no idea what the cost will be, that will have to come from the nearest repair center as the damage may have spread from the heat. Sorry to be the bearer of bad news.