This sounds like two different hardware malfunctions.
1. display:
There are several things that might cause the problem. Since an external monitor works we can exclude the graphics card itself. Maybe it's the backlight that is broken. You can verify this with a flashlight illuminating the display from a flat angle. Then you should slightly see the desktop.
If it's not the backlight most likely it's the inverter. But there is also a little chance that it's the panel itself that is broken. I don't know how to distinguish which of both causes the problem without advanced electrical tools.
Whatever the problem might be, fixing it requires some experience (or daringness
).
2. battery:
Either the battery itself or the charging control is broken. If the first is the case replacing the battery with a new one will solve this. If the latter is the case, forget it. There is no reasonable way to fix this.