To output 1080p content at 144Hz, you typically use a dual link DVI-D cable or DisplayPort cable. You can also use a HDMI 1.3 or higher cable.
Running 1080p content at 144Hz can be a very demanding task. Contrary to popular belief, a standard HDMI 1.2 cable can not be used to output 144Hz 1080p content on a monitor. HDMI versions up to 1.2 support the same data throughput as single link DVI-D. A HDMI to DVI or DVI to HDMI adapter is converting to or from single link DVI-D, as the video signal for these two is identical and it is just the adapter that is required (HDMI can carry audio, but this is lost with an adapter).
This is sufficient for 1080p at 60Hz.
To output 1080p at 120Hz or 144Hz, you need a dual link DVI-D or DisplayPort cable and connect those to your monitor and computer. These standards include support for higher resolutions and refresh rates. HDMI 1.3 or higher can also be used to output 1080p content at 144Hz. If you need a 144Hz monitor then be sure
If you need a 144Hz monitor then check out or list of the best gaming monitors as it includes all sorts of 144Hz monitors.