A very close question was discussed on MathGroup in the following thread: Easy way to read clipboard?
You need to click a few times "Next by thread
" link under the post to see the replies. There was no dynamic automatic updating though. Based on those answers I suggest the following:
Dynamic[Refresh[NotebookGet[ClipboardNotebook][[1, 1, 1, 1]], UpdateInterval -> 0]]
But beware of complications. If you copy simple things - like strings, numbers, variables, etc. - this will work just fine. Nevertheless a clipboard can contain much more complicated expressions, - multiple cells for example. Then the resulting structure needs to be more carefully addressed than the trivial Part extraction [[1, 1, 1, 1]].