My 6115CD, very similar to yours, had a finicky monitor cord. That might the problem with yours too. Make sure that you are careful about plugging the monitor in, but make sure you get it firmly in place.
The battery is not hard to replace and yes you'll have to open the computer to get to it. I don't recall exactly where the battery is located, but it's easy to spot. There's a plastic thing snapped on top of it, if I remember correctly. If not, you'll see it easily enough. Like I said, the battery looks like a cut off AA battery.
Try securing the monitor plug first. If you still get the black screen, then try the battery. The battery is one of the easiest things to replace and by far the cheapest!