Your phone is not on WiFi if the screen is off, unless the phone is plugged in. WiFi turns off 30 seconds after the phone goes to sleep to save battery energy. It will stay on when plugged in. The phone can run many apps when it is asleep; the built in apps all update continuously, and 3rd party apps are by default permitted to run in background when the phone is asleep. You can allow or deny this capability on an app by app basis in the Settings app.
The simple solution is to leave the phone plugged in overnight. It won't hurt the battery, as the phone is smart enough to stop charging when the battery is full. It has the added benefit of guaranteeing that you will have a full charge at the start of the day.