i dont know why but IPhone 5S was suppose to be the best device of apple. According to apple, the device is capable of multitasking, high end gamming, and efficient browsing. The device is equiped with A7 AND M7 chip, thou is hopeless. The deivce laggs a lot, freezes at times, battery is draining rapidly.
There is something running in the background on your device. Most likely improperly. It's hung up on some kind of command...
If you are/were restoring from an iCloud backup, it may be stuck on that. (Those restores can only be done on WiFi and can take 1 to 4 hours per GB of data coming down)
If no restoration is taking place, try this...
Settings/General/Reset/Reset All Settings
This will not erase any data, it will simply reset the temporary preference files the internal CPU uses and go back to factory defaults. So you'll see your background go back to the spacey looking one, and your wifi passwords will have to be put back in... Siri and Location Services will be off. All of this stuff is easily repopulated, and it will most likely fix your problem.
There are deeper issues that sometimes require a full OS restore and setup as new device.. but usually thats a last resort.
You can also make a genius bar appointment at an Apple Store where they can run some diagnostics to see exactly what the hangup is. It will also check some hardware, but 9 times out of 10, this is a software issue.
The answer to the rest of your question is in my original post. Reset all settings "should" fix whatever is heating and draining your battery. (The heat is coming from the processor which is in turn draining your battery) Now if you've done the reset all settings, charged it to 100%, and then 4 hours of "normal use" later it's dead, you may have a deeper software issue that will require a erase and setup as new, or you're having an actual hardware issue that will require either a Genius Bar appointment or calling Apple Care directly. The diagnostics they run gives you a detailed report about battery drainage and what is causing it.
There can only be 3 things causing it,.. Software, hardware or usage. That will definitely answer your question.
I emphasized "normal usage" meaning web browsing, email, phone calls, light gaming. If you're watching a 3 hour moving streaming over 3G/4G/LTE then you're going to see a huge drain. It's good hardware, but not magic. Haha. I doubt that's what you're doing, but just in case here's a support article that you can reference. http://support.apple.com/kb/HT1805
If you can not fix it by these methods, I strongly encourage contacting Apple either in person or on the phone. That will give you the most direct answer. You have a 5s, so part of your warranty is that support.