My spec is i5 8600k, 16GB RAM, GTX 1060 6GB and I’m using an 1440p monitor. Since I’m using a fix renderscale of 75% it means that the rendering is done basically in 1080p like in your case.
In order to achieve stable 120fps I’m not running with max (only high) graphics settings but it looks wonderful. I could upgrade to a GTX 1080 but I don’t play a lot of different games and OW looks good enough with my current setup so that I didn’t bother with an upgrade.
On the login screen the game uses the in-game fps while in the main menus it always caps the fps to 60 (because it would make no sense to stress the cpu when you aren’t in action, 60 is very good for clicking menus). On the hero selection screen the GPU usage is a bit higher than during normal gameplay (probably because of the blur effect on the background). For this reason I recommend tweaking the video settings by starting a custom game.
In another tech support topic I gave a very detailed explanation of how I tweaked my video settings and some of those tips might help you to debug your issues too: Performance Problem Do I have a bottleneck? - #4 by TRON-2667
You might be able to use msi AfterBurner or similar software to print useful info (CPU/GPU usage) to the screen while you are in-game.