I would bet that your alternator is not putting out... (Like my ex-wife)...
The battery is only supposed to start the car, after the engine is started the alternator takes over to supply the power to run the ignition and accessories and replace the charge in the battery that you used to start the car...
If your alternator is not putting out any charge, the car gets it from the battery... You are then running off the battery until it runs out of charge and will die...
If you put a charger on the battery and started the car and it ran, then died when you disconnected the charger, chances are the alternator is not putting out enough charge and you are running off the battery until it dies....
Try checking the voltage on the system when the car is running... A good charging system should run between 13.3 - 15.0 volts... Any less and your alternator is not putting out enough, and any more the voltage regulator is not doing it's job....
Fully charge your battery and then start it and disconnect the battery charger and check the voltage across the battery terminals...
If your battery will not take a full charge, then your battery may be bad also....