To effectively take screen shots, you will need to work in the dark, or near-dark. This means you need a tripod. The sturdier and heavier, the better. A strong, heavy tripod won't flex and it will not be bumped as easily. This is important because the trick to screen shots is having them all be positioned exactly the same.
If you have 20 shots, you want them to be identical in framing. That way you can define a Photoshop macro or something to automate the processing of them. Do it once manually while recording the macro, then run the macro on the rest. Everything will look the same. In theory.
Of course you need to get them in focus, and a non-SLR camera will be a bit hard to use in that regard. But do what you can. Try to focus with the lights on, then turn the lights off to get the shots.
If your camera allows for manual setting of the light temperature (known as "white balance"), most displays are rated at around 6000K, so use that as a starting point. If in the photos the screen looks too blue, raise the White Balance temperature. If it looks too red, lower it.
That's it. I'll add more if there are other topics I should cover. Let me know.
Try to get the camera