I know that higher versions of HDMI can do a higher refresh rate at 1080p, but do I just need a higher version HDMI cord to get more than 60 Hz or do I have to make sure that the monitor supports over 60 Hz via HDMI as well?
HDMI does not support monitor overclocking since it does not have high enough bandwidth. Monitor overclocking will only work with dual link dvi cables and display port cables.