The first two replies are misinterpreting the problem. The OP wants to use the wires in a cat 5 cable as a HDMI cable. Wires are wires, so it should work as long as there are not impedance/crosstalk/twisting/shielding issues that make the signals transmitted incompatible. Otherwise the cable doesn't care what kind of data it transmits.
The above adapter requires Cat6a shielded (STP) cable for runs between 30-50 meters. It's probably a good idea to use that kind of cable even for shorter runs if possible, for best results.
Monoprice sells Cat6a STP cables for a reasonable price
http://www.monoprice.com/products/p...=10232&cs_id=1023216&p_id=5905&seq=1&format=2
(also in a variety of other lengths).
edit: I though I'd add that HD over the internet is streamed highly compressed and requires very complex encoding and somewhat complex decoding. HDMI sends data uncompressed, so instead of megabytes per second it requires gigabytes per second (up to 10gbps for the latest HDMI spec at the highest resolution).
For extremely long distances (more then the length allowed that adapter), fiber optic transmission would probably be required. But once you go to fiber, there is virtually no length limit.