I am experience this issue since a couple of weeks.
My laptop needs to be plugged so it can start windows.
Hard disc is new and battery. once its working it works well on battery.
But every time i restart it, screen goes black and comes alive once cable is plugged in the wall.
Save up you money to buy a new laptop. This strange problem is probably caused by a hardware fault in the motherboard power circuits. Most motherboard problems call for replacement of the entire motherboard and cost far more that a simple replacement of the entire laptop.