This is known as a RAT (remote access tool), I used to build them into my apps for ease of support.
Look here for code on capturing bitmaps / screens (I have posted this code before). It will show you how to get the current image on the sceen and convert it to a bitmap.
Then you only have the problem of getting that code to run on the other machine, connect and send the bitmap to your machine and display the bitmap (which is against the rules on this site).