Looks good, but if I were you I might change the R5 3600 to a 2600, maybe even used and a used b450 mainboard for out-of-the-box compability without bios updates. The money saved probably arround 100 dollars should be enough to bump you up to a 1660ti or even better a RX 5700. Maybe you could even aford a bigger ssd.
Edit: https://pcpartpicker.com/list/kmpBb8
I did not include a case, because you havent. Better psu, bigger ssd and a 1660ti. I would go used on the mobo and cpu to save some more money, but thats an individual decision.
If you do not want the case yot have in the build, because you want something more expensive, then just get 8gb of ram instead, so you can buy a more expensive case that's on budget.
There is not best case, but I think(and probably others too) this is the best looking case under your budget. All you need is one 3.5 inch bay in this build, or if you want to use an optical drive, than you can find plenty of cheaper cases with a 5.25 inch bay(s).
It has 16gb of DDr4, 512gb of ssd storage + 2tb of hd storage, even a cpu waternlock, and best of all...... IT HAS THE LEGENDARY LIMITED EDITION i7-8086k!