HDTV is basically a tv that supports higher resolution. A monitor with 1280x780 is 780p, and a monitor that supports 1920x1080 is 1080p. Don't look for a monitor to say HDTV, because its not considered a TV, and monitors have been able to exceed regular TV definition for a while.
To tell how high your resolution on your monitor can go, is by right clicking on the desktop and going to properties, and go into the Setting tab. Slide the Screen Resolution slider all the way to the right, and take note of the resolution (800x600, 1024x768, 1280x1024, etc). Anything more than 640x480 (480p) is better than your standard tv, and the xbox 360 will allow you to choose which resolution you want. Just make sure you don't exceed what the monitor can handle.
EDIT: As for Kameo not doing the Xbox 360 Justice, I think most people are confusing its art style. I mean the amount of detail on the scenery are amazing.
XBox/XBox 360 Comparison
EDIT: Didn't notice the link to your monitor. Yeah, your monitor could support 1240x1080, thats about a 780p (actually a bit better).
XBox 360 running on a Dell Monitor