Now this will probably be a short update. First I'll mention the easiest to clean, but now is mostly obsolete with all companies. Glass screen monitors are quite easy as you can clean them the same way you clean a window. The only concern is that you do not get liquids into the monitor (spraying into the container of the monitor).
LCD monitors are a lot more difficult to clean since they are very sensitive to getting scratched and or getting damaged by there being to much pressure applied when cleaning. It is best when cleaning to turn off the monitor just so it is easier to see where it is dirty. The first thing you want to do is get a specialized cloth which I believe is called a microfiber cloth (the same cloth used to clean sight corrective glasses) these cloths are designed specifically not to scratch sensitive surfaces. When using the cloth DO NOT put too much pressure on the screen because this can damage the pixels on the monitor. If you are unable to get a good clean it is okay to put purified water onto the cloth. But DO NOT spray or put water directly onto the screen as it can damage the interior of the screen.
Now as a side note, laptops... they are difficult to work with since they are designed to not be changed in the interior due to the fact that they are made to be small. Most problems like overheating or anything having to do with the computers parts are not very fixable. In most cases you will have to go to where you bought it or contact the laptops manufacturer to get parts fixed. They are just designed to be small and efficient and in that they made them nearly impossible to work on personally.
Yet again if there are any other questions just leave a comment.