It seems like you are talking about two separate problems. Your first problem on how your phone does not display a charging symbol when plugged in could come from your phone charger or your phone's charging port. Try using a different charger, if that solves your problem your last charger is no longer working. If a new charger did not help your problem you will need to replace your charging port on your phone.
For second problem with how your phone dies immediately after unplugging it from your charger is due to a faulty battery. There are no solutions to this other than replacing your phone's battery.