Your post is a bit confusing.
If you're wanting to connect from your deck to your computer, you should just use some analog stereo RCA cables. If you're going to be recording records and cassetes you're going to have some background noise always there, and I wouldn't worry too much about buying high-end cables.
No Hdmi to DVI involved here. DVI doesn't carry audio signals, as far as I know.
Good luck!