The options for what you want really depend on your setup.
If your monitor can accept accept component input, you can use the HD AV cables; use the components for the monitor, and use the composite (single yellow) for capturing video to your PC - you'll need either a video card or a specialised video capture card / TV tuner card with composite /input/. You'd probably need a dual female RCA plug to 1/8" male stereo plug adapter. You might need to split the audio to go to both the computer and the monitor, depending on if you have sound card that can output the input from your line-in in real time. (Sorry this all sounds complex, it is)
If all you can use on your monitor is DVI / VGA, you're going to want to use an entirely different setup. Now, this setup will cost more, but it'll actually give you better picture quality. It can also be used instead of the above, and is probably a better bet for usefulness over time. Basically, you'll want to get an Xbox360 VGA cable, and then run it into a device that does conversion to S/Video but also has a VGA passthrough. You'll also need a Video Card / TV Tuner / Capture Card with S-video input; it's what you'll use for capturing. In this setup, you get VGA input to your LCD, and S-video for your capture; really the best you can do right now for a small budget (there are no cheap VGA/HD capture cards). Unfortunately, the major drawback of this plan is /finding/ the type of device I just described - they're not exactly common! eBay might be your best bet.
If you can't do either of those, your only real option is to simply get a Video Card / TV Tuner / Capture Card with S-Video input, run the 360 into it, and use a program like DScaler for playback and recording (it can do both at the same time, provided you don't save your videos with a CPU-eating codec). Quality will be low - your 360 will be outputting at 480i (interlaced, not HDTV), and then your PC will have to deinterlace it. This is the route that I use, sice I have CRTs, and I'm not willing to use the 360 VGA cable and game at 60hz on a CRT. Games still look nice, of course, but they're low resolution, and tweaking the brightness/saturation/etc settings to be just right is a pain.
