It's computer compatable, you will need to buy a cable. The TV has every type input. I suggest the DVI/HDMI. Your specs, could not find via your link: http://32le5300.com/
The hdmi will display the full 1920x1080 through the pc? I see that they advertise the D-sub as the pc input, and im pretty sure they use wxga which is 1366x768 instead of wuxga which is 1920x1200/ 1920x1080
it should as long as you specify your screens resolution through your computer. one option, make sure (before you purchase the TV) that your graphics card can handle that kind of resolution.
Using the HDMI output of the graphics card (as mhelm1 recommends), you should be able to select either 1080p or 1920x1080 in Catalyst Control Center as your default resolution.
hdmi to hdmi, or dvi to hdmi? (video only)
DVI connector on your 5870 to one of the HDMI connectors on your tv. Now you have to listen to sound from the computer speakers or set up the computer to enable audio thru the HDMI-DVI. Or connect audio fron computer to your TV. More info here: http://en.wikipedia.org/wiki/Digital_Visual_Interface