You need to buy a few parts before that works. I did some research on this a few years ago - the problem is that the PS2's video output is interlaced, you'll have to buy some adapters to get it to work on a VGA connector. There are some expensive ($40 range, but maybe they are less expensive now) devices that will do all the work, or you can google around and scrounge around some parts to put everything together yourself.
That said, I never did pursue hooking my PS2 up to my computer monitor.
If you use a component (blue, red, green) cable instead of the composite (that comes with the PS2), the picture quality is a lot better, especially if the game allows you to use progressive scan.