Can you change the iPhone 5S battery ? Yes.
How Much Does It Cost?
Through Apple: Battery replacement for an iPhone 5s is $79 plus whatever tax if you're out of warranty. If you're in warranty and Apple deems it necessary, you don't have to pay.
If you're not worried about support coverage, You can also buy the original battery online from amazon, iphone-battery.com and do it yourself.
I would at least first get the phone assessed by Apple to see what they deem necessary. That much is free if you can get to an Apple store. They may agree that a battery replacement is what is needed, or they may have a different assessment.