I'm not familiar enough with Jing to know exactly what it uses. But there are two basic techniques. One is as you mention, capture the screen and display it in a topmost borderless form. The Vista/Win7 Snipping tool works that way. You'll find the code you need to get this started it in my answer in this thread.
The other, perhaps more likely to be used by Jing, is similar to what Spy++ does, allowing the user to move the mouse and draw a selection rectangle around the window. Its advantage is that it can deal with windows resizing or disappearing while you've got the tool running. You implement it by using a topmost form the size of the screen that has its TransparencyKey property set to the value of the BackColor. Fuchsia is a popular choice. You can draw on this form with the OnPaint() method, the drawing appears on top of all the windows. You'd need some P/Invoke (GetWindow) to iterate the underlying windows in their Z-order to know which window the user is pointing at. GetWindowRect() to get the window rectangle. Plus some hassle to deal with Aero lying about the border size.
You can find sample code to get you started on that technique in my answer in this thread.