Hi,
1) It very rarely makes sense to game at 4K anyway. You can run most games at 2560x1440 and have it scaled to 4K on the monitor with similar visual fidelity to native 4K.
2) GPU. Yes, this is usually the limiting factor, and unfortunately that may mean buying TWO top-end cards (like 2x GTX1080 in SLI). However, not all games support this and when they do there may be stuttering introduced.
*So unfortunately, there are people with two high-end graphics cards that have a LESS enjoyable experience in some games because they got the second card. They got an arguably nearly IDENTICAL visual experience in terms of the quality of each frame but again stuttering (game not smooth).
DX12 and Vulkan games will eventually use multiple GPU's properly (to both work on the same frame, not one per frame), however that probably won't start being common for at least two years. That needs to be baked into the game engine but it's a difficult problem especially whilst maintaining backwards compatibility with DX11. So I recommend a SINGLE GRAPHICS CARD.
3) 4K HDTV and 4K monitor is not the same BTW.
AFAIK with an HDTV you can't choose a resolution like 2560x1440. So you're forced to run at 1080p or 4K. That almost always will be 1080p as game smoothness is more important. Plus, if you aren't sitting really close the game might not even benefit from 4K resolution.
4) TWEAKING the game settings goes a long way too. Often there are settings which are really taxing that don't add a great deal visually. You have to experiment.
You should learn how to use Adaptive VSYNC as well. It turns VSYNC ON and OFF automatically. On 60Hz monitor aim for minimal (i.e. 5% of time) drops below 60FPS. When it does drop you get some screen tearing but not added STUTTER due to the sync mismatch by having VSYNC ON but not being able to keep up.
5) GSYNC - for desktop monitors, GSYNC is a bit expensive but also provides a smooth experience. The monitor only draws a new frame when the GPU sends it one rather than updating at exact intervals (i.e. 60x per second). This new, asynchronous method prevents screen tearing, reduces lag and prevents stuttering (VSYNC type of stuttering at least).
Example monitor: GSYNC, 144Hz, 27", IPS ($800USD?)
For a shooter I might aim for about 100FPS average. Some games need a 60FPS cap.
A good example of a PC for that would be:
i7-7700K (or RYZEN 4C/8T CPU)
16GB (2x8GB) 2666MHz DDR4
GTX1080 (or similar AMD Vega GPU later)
etc.