I have a video player connected to an ATI graphics card and often copy old home movies to HDD.
Obviously you need the software to capture the images (Studio 9).
As for the quality every copy will reduce the quality of the image but I don't think that its noticeable. Once copied the .avi files are quite big and a bit "blocky" but once authored for DVD they are fine.
Connecting PC to TV seems more trouble than its worth - I suggest a cheap DVD player (maybe 40 quid) is a lot less hassle.