It may be simply down to wear and tear of the battery; an idea of how old it was would have been useful.
Nevertheless if it's become a major issue take it into the apple store (best to book an appointment as they are usually very full).
Otherwise, I reccomend turning off Data when your not using it, push, time zone settings and location features on apps you don't really need it for.
Phone not holding a charge even when no apps are running in the background.
It is impossible to have no apps running in background; the built in apps will run in background even if you kill them; they restart automatically. So, you either have an app using power or a bad battery. If you take it to a genius bar they can test the battery, which will narrow the problem down.