I can get 1920*1080 without drama on a Sony across a DVI to HDMI cable. A HDMI cable is really just a DVI cable with a fancy plug. The digital signal for DVI and DHMI is identical and since it is a digital signal, cable quality is immaterial as long as you are getting a signal.
You really can't improve the resolution of your screen beyond its' native resolution. If you have a LCD screen with a native resolution of 1360 x 768 then that all your going to get.

