Assuming you're talking about the Xbox 360 Elite, and that you have a monitor that has HDMI inputs... I would go with HDMI, since HDMI is basically DVI + Audio + signal encrypting copy protection garbage. It's the same deal as comparing VGA and DVI image quality with a computer. DVI being digital end to end avoids the quality loss from signal conversion that VGA suffers.
Though you may well be disappointed using a computer monitor and HDMI if you're looking to use higher resolutions. For the longest time, computer monitors have been vastly superior to television sets, but with some of the latest HDTV sets, television has taken back the lead. Few, if any, computer monitors can do 1080i or 1080p at present. Even 720p is pretty uncommon. Most can really only scale 480i to something close to their native resolution.
will the picture quality look better on a computer monitor via: VGA hook up or using the HDMI hook-up with the elite package?

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