If you are no competitive gamer then a TV can be fantastic bang for buck, but it does have its negatives.
You have to be quite specific when trying to buy a tv for pc gaming because the specs of the tv really will make an overall difference in many scenarios.
As Orcvader says, 4k gaming is where you really will benefit from those real high end cards such as the 1080/1080ti, the processor can also keep up as the gpu will have to shift 4x the pixel density per frame compared to 1080p. Using 4k instead of 1080p will roughly quarter the framerate output so the cpu can process the lesser frames more easily.
A good example: 1920x1080 resolution gaming with a 1080ti may produce 300 frames per second for the cpu to then have to process (bringing its knees) where as that same gpu being pushed at 3840x2160 (4k) will create roughly 4x times the picture quality with 4x the pixel density (meaning it's working 4x harder thus creating 4x less frames for the cpu to have to process and at a far more reasonable 75ish fps).
Some people prefer framerates to eye candy (people like smoother gameplay with no shuddering/ghosting in fast motion scenes), personally, i prefer eye candy as long as the frames are enough to barely notice this shuddering/ghosting effect....i think you can really feel it/see it with the game being not so enjoyable when its being played at 30fps or lower. The best i find to aim for (and what is deemed to be the minimum benchmark) is a solid 60fps. If i can add eye candy, id happily sacrifice framerates above 60fps to add some more......
This is where you need to be picky about tv specs, it means your TV needs to be able to cope with 4k/60fps, so you want your tv to have a 60hz panel, but also be aware that some tvs (admittedly not so much with the newer ones/decent branded ones) can have a 60hz 4k panel but only have old hdmi revision ports on their AV boards. These tvs can do 60fps but only at 1080p and although the panel can technically do 60hz 4k, the hdmi 1.4 revision ports on the av boards are restricted to only supporting data bandwidth for 30hz (30fps) when plugged into a 4k source. Make sure the tv has a 60hz panel and at least an hdmi 2.0 revision port which was the revision to add support for 60fps at 4k with hdcp 2.2 (hdmi 2.0a further adds support for HDR content and the newer 2017 hdmi 2.0b ports add support for hybrid log gamma). Whilst i'm on the subject it also reminds me of another total waste of money on bragging rights....people buy a PS4 Pro for gaming and brag it delivers HDR content but most are using non hdmi 2.0a revision TVs, this in turn renders the HDR feature on the ps4 pro absolutely useless.
Anyway, back to subject....so yes..... a tv with hdmi 2.0 (at least), a 4k display with a 60hz panel and finally you'll need to check reviews or specs for input lag.....Unlike monitors, this is where tv's lack and have their downsides....most tvs have a bit of a lengthy input lag, this is where you will say press to say fire a gun and the response on the screen will come a split second or so after (due to the tvs image processing).....some can be really bad in this area where as others can be pretty good (plenty low enough lag to barely notice)....My tv has a specific games mode which reduces input lag to around 10ms, this to me is barely noticeable (some may argue otherwise) but i have seen some (and used some) that when you press fire for example, it is almost a second or 2 delay before you see the action replicated on screen.....you really don't want this.
Some tvs offer better and faster processing but a lot of the main manufacturers use funny terms for their panels and processing (such as clear motion index or clear motion rate)....just try to get as faster processing tv you can afford with a 4k/60 panel and at least hdmi 2.0 ports.... that way youll be set for 4k gaming.
As i myself prefer eye candy over framerates (as long as near 60fps) i persoanally use a 65" 4k HDR samsung TV as my main driver. I'm also no keyboard fan for gaming so use a PS4 controller and when i'm browsing or doing standard work related tasks, i use a standard, lightweight, non flashy wireless logitech keyboard with a built in trackpad/touchpad (like a laptop)....this way i can cabbage myself on the sofa and browse in comfort, Sod all these stiff upright car seats with your eyeballs pressed against a monitor so small there's so much density of pixels you don't need more than 1080p... i'd rather be laid back on a sofa, in comfort and see the crispness from a distance on a huge display that's cost less than some 32" monitors do (whilst also justifying and seeing the benefits of those 3840x2160 pixels you've paid for).
To finish up this huge enthusiastic Essay (i'm a tech geek), Orcvader was also right when he states about 4k video on processors... displaying 4k video that's already been developed (so to speak) is easy for a processor, it's simply read and display, i'ts when you ask a cpu to render or create 4k moving images from scratch/code that it becomes taxing. Be aware however that as far as i'm aware, if you happen to use the very common Netflix 4k and intend to use it on pc that any 6th gen intel cpu or lower wont support 4k from netflix. Netflix requires 10bit HEVC decoding support for its codec and this was not added to intel cpus until the 7th gen kabylake lineup,. (another potential reason to get the 7700k).
Wow, this has been a one hell of a big write-up....i'm far too enthusiastic..... once again i hope my rambling on has made some parts sense, good luck with it all, if you need anymore help, give me a shout and i'll try and do my best to help.
. You might've got more worth for your budget.