Most of your high end TVs have a native 120Hz refresh rate. The Samsung Q8 and Q9 series even have freesync with it.
The issue is you are going to be looking at easily $1k for a TV with that refresh rate, I believe the Vizio P series has a 120Hz rate, and the Samsung's start at $2k. You would probably be able to buy a good monitor for less money at that refresh rate.
What type of games are you trying to play?
competitive fps or AAA single player or what?
For competitive games if you want the best don't buy a TV. size is relative to distance. sitting at a desk a 24 or 27 inch monitor is fine. a lot of console players even do this. 24inch 1080p 144hz 1ms monitors run about $200. you have one of the best cards so you could buy one of the 240hz monitors and play a lot of games at that.
For eye candy games it's honestly best to just buy another monitor. TN monitors colors are pretty bad. There aren't a lot of options that are best of both worlds. There are some newer low latency non-TN monitors that will have better color, but for FPS most go with TN.