This isn't a job for Chrome cast.
You just want a long DVI / HDMI cable really, it's a very simple setup. DVI will need the usual sound considerations, but otherwise, it's an easy one.
No special cable needed, so the Scan one should be fine. If it's not, it's faulty and you can send it back. The beauty of digital
edit - one small thing. Make sure your DVI port can handle HDCP over it. Most can these days, but older ones can't. If it can't you'll have issues with protected content (if played through a PC) like Blu-rays.