Try charging it when the laptop is turned off and see if it will charge then. If yes, see if it starts up then once it is charged. If it will, then the charger may be your problem and not putting out enough power to both run the device and charge the battery.
If it won't charge when off either, or it won't turn on after charged up, then the battery may be a faulty one. Or the wrong specs. That can happen if it is an after market battery. If that is the problem, then I would return the battery and get a better one.