iPhone 5s phone - Battery die fast and power off auto and ask for to connect charger

I have iPhone 5s phone, had battery issue. Initially I thought it was with battery but it's not. i replaced with new one but still the same issue. I charged to 100% but when I leave it for night the phone would be dead in the morning and asked to connect charger. The battery is dead without even using it. Sometime I feels phone is really hot when it's being charged. Another noticeable thing is; when we left shutting down the phone battery won't go down, it's good but that doesn't make sense right, we can left switching off the phone. Please suggest, anybody?