Anytime you view a YouTube vid it is already in the cache of your browser, so there is no need to d/l it directly once it's fully loaded.
If you can't find the cache or don't feel like navigating through your files, there is a program called VideoCacheView v1.71, it scans your caches and gives the option to copy the video file in your cache to whatever folder you want. Play offline/Save .flv video files from Web browser cache
Once you have the video, you can use something like DVD Flick to convert and burn it to a DVD. You'll find it here along with a list of other free video tools DVDVideoSoft.com: Download free DVD, iPod, YouTube video converter and editor software and freeware
*Note - DVD Flick runs best when set to compatability for Win XP Serv Pack 3