If you want to do this, you have to buy a USB TV Tuner and plug your PS2 Video and Audio into that, or you could install some sort of graphics card into your computer or w/e that has video in and audio in (like TV's do.) Other then that, there really isn't a way to do it last time I checked. and screw PS2 Emulators.
Once you've captured the video you MAY be able to edit it using dvd-shrink by using the remove start & end frames action it is possable to remove & save the sections you want or if you need to change the encoding so shrink can read it mediacoder may work or not,have fun