iPhone 5 Battery Won't Hold Charge

I got an iPhone 5 last Feb. I had no trouble until lately. The phone doesn't seem to hold its charge. A fully-charged phone has only 45% battery strength the next morning with no usage, is very strange. I am forced to charge 2 or 3X daily. I also notice the white block which gets plugged into the outlet gets extremely hot during charging. Is this normal? Another phenomenon, whenever the phone is plugged into this PC, it automatically charges. Where is the power coming from? Is it taking power from the PC? If the battery must be replaced, what is the cost and won't I lose all data when it's returned? Thanks.

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I would take the phone to Apple. It should still be under warranty, right?