The main difference between a TV and a monitor, besides the TV tuner, is the pixel density. A Full HD-TV has a resolution of 1920x1080, which is similar to common monitors, but spread over a significantly larger area. The consequence of that is that the recommended viewing distance for a TV is much further away than for a monitor.
In short, if you want to sit on a couch and play, get a TV. If you sit on a desk, get a computer monitor.
For a Kinect a relatively large distance to the screen is required, I would recommend a TV for that reason.
All devices that you have plugged in should show in the Windows Device Manager, whether they have a driver or not. In Windows 7, hit the windows key and type device m in the search and you should see device manager, select that.
In the top level of the device manager, if you plug in the wireless controller, you should see a top level of "Microsoft Common Controller For Windows Class" if you don't see that, and the device is actually working, it should show in an "Other devices" or "Unknown devices" section. If that's the case follow the instructions in this article. Or better yet, this article here which seems to work for people that can't get it going any other way. It says it's for Vista but Vista and Windows 7 share the same driver model.
If you plug it in and nothing happens at all, then you probably have a dead device. I believe the light comes on when it gets power, even before the proper driver is loaded. (not 100% on this) Try it on a second computer.
I had a perfectly good Xbox 360 wireless adapter that worked for a few years, not tons of use. Then I pulled it out of a drawer after it lay dormant for 6 months or so, and plugged into a computer it worked on before, it didn't do anything. I tried a second computer, same thing. The computers didn't even recognize it as a device.
I went out and bought a new one at Best Buy, plugged it in, and it worked, no problem.
Best Answer
Yes it will work on your PC (provided it has a USB port).
There is also an open source "SDK" with reverse engineered drivers for Windows and Linux, if the licence for the official SDK is to prohibitive (there are a lot of licence restrictions on the use of the official SDK, a commercial SDK might be released later on).