Car won't start, dies immediately after removing jump start cable

[Jeffery]

Hello everyone needing some desperate help!

I have a 2002 Toyota Corolla Runx Z and I was installing an aftermarket intake. I drove it into the driveway that morning and started getting to work only to find out that my dash lights were off even when the key was set to the on position. Upon trying to start the car, it wouldn't start and I get this humming/buzzing noise near the passenger window (definitely not turning/cranking sound of solenoid). I tried jumpstarting the car and it would start up just fine. But after letting it idle for a while, when i go to remove the jumper cables the car would die immediately. Any idea what might be the issue? I am assuming it is the alternator, as I have had a battery light on (extremely dim) since buying the vehicle, however never had problems with the car/battery/starter etc. until I removed the battery. 

Any help would be much appreciated ! 

[campbeam]

I am thinking that you have a possible combination of a very discharged battery and an alternator problem.

The engine should run with a fully charged battery.

Battery warning light being displayed indicates a battery charging issue i.e. alternator.