Digital cable on second monitor

So I built a new system and one of the things I wanted to do was watch cable, high def, on my 2nd monitor. So I got a tv tuner card and hooked it up. I ran a splitter from the cable box but I only get basic cable and no high def. I moved to the office my wife and I created out of an empty bedroom. I got a new cable box from time Warner and was all excited. I hooked everything up and realized that the output on the box is only HDMI or component, no coaxial which is what the tuner card has. So I was thinking about it and I realized that I could run HDMI out to my monitor. The monitor has a audio out and my motherboard has audio in. I am just curious if this would work? I won't be able to record on the computer or use the Monitor as a second screen but at least I'd have tv.

As a side note, anyone know of another way to get full HD cable, all 900000 channels on the computer?

Yeah you can- if your monitor supports it, support being the key word here.

it depends on the monitor, but most monitors are above 720p, while most 'HD' cable providers are only on 720p, so it will work, but it may look worse on the monitor then it would on a TV. TV's are glorious because of their LACK of quality. They tend to blur things over which makes most low quality things look better than they actually are. A monitor tries to be as exact as possible... so when you have a crappy feed coming to it then it really stands out. Monitors also lack the contrast that a TV has, so it does not have that same 'wow' factor that draws you in.
To every thing a purpose, and to every purpose a tool. TVs are great at TV, while monitors are great at computers, and while there is ever increasing overlap, there is still a distinction between the two.