I can't figure out why my battery keeps dieing.
Battery light on when car is OFF
Battery light doesn't light up when key to ON position
Car runs, like a beast, for 5 minutes before dieing from dead battery
So after you charge the battery it'll run for a short period of time then dies?
If so, your alternator is bad, once started the car pretty much runs off the alt. shouldnt die for a dead battery. Hence why roll starting will work if your battery dies and your stuck somewhere.
Could possibly be a bad alt cable, a blown fuse in the alt cable, a bad connection of the alt cable, check these things and if it still doesnt work take the alt anf have it tested.