Replacing the battery should fix the discharge problem. Luckily, ifixit sells it.
[product|IF250-014]
However, if it doesn't charge well even after the battery replacement, your charging port/cable is bad or your charger isn't powerful enough. First, get a brand new Samsung cable. Not a flimsy one you would get at a convenience store. Also, make sure your charger is at least 2.1A 5v. If neither of those fix it, you'll have to replace the charge port.
Hope this helps!
[product|IF286-000]