A week or so ago I got an iPhone 5 16GB. In this week, the battery runs out stupidly quick (within two hours, even if I don't use it at all or it is shut off completely). At first, if I had Wi-Fi, the battery would act semi-normal, but now it dies quickly even with Wi-Fi. And today I hopped on an app for a minute or two while it was charging and the battery dropped 25%, while it was charging. What do I do? I literally just got this phone a week ago.
It could be an app that is using all the juice, but more likely you got a used iPhone 5 with a bad battery. Where did you buy the phone? If it is from a reputable business, take it back. If you did e-bay, craigslist or something similar you are probably out of luck. You can take the phone to Apple to get a new battery put in for a set price of $79. You may get cheaper at a local iPhone repair center or kiosk in a mall.
I would also check here to see if your iPhone 5 battery is on the recall list: