Honestly, if I was in that position, I'd just do it piece by piece and focus on one thing at a time. You usually try to pair up the display with the GPU so one isn't too far ahead in terms of spec for the other. If you try to get little of everything, you're going to spread out your funds too thin and none of the stuff you get will be worthwhile
As you can barely build a decent gaming rig that will last more than a year or two with the current budget, there's no way you'll fit in a display on top of it. I say get a relatively new i5, throw in a GTX 1060, lower resolution/quality for the 4K TV and play as is with whatever input lag that already exists.
When you have another $500 saved up, you can get a GTX 1080 (or better by then) to be able to bump up the settings on that 4K TV. Alternatively, you can save up $200 for a decent 1080p or 1440p display with higher than 75Hz refersh, preferably a IPS variant for better color reproduction and good response time. Ultrawide screens are very popular nowadays, so maybe look at some of the 29" LG offerings if you like good, affordable ultrawides.
edit-
Here's a decent one that's currently on sale:
https://www.newegg.com/Product/Produ...82E16824025038