Luckily the Cryptomining craze is dying down and you can find mid range GPU again. At least you can find a mid range nVidia again. AMD prices are still obscene. nVidia prices are still high but not too much.
The GeForce GTX 1060 6GB is the best you can get within you budget. It will give you the most performance.
https://www.newegg.com/Product/Product.aspx?Item=N82E16814125879
As you want a case and GPU. You'll have to settle with a GTX 1060 3GB. The NZXT S340 is a popular case. It is quite nice and offers good cooling.
PCPartPicker part list /
Price breakdown by merchant
Video Card: Gigabyte - GeForce GTX 1060 3GB 3GB Windforce OC Video Card ($224.98 @ Newegg)
Case: NZXT - S340 (Black/Red) ATX Mid Tower Case ($69.99 @ Newegg)
Total: $294.97
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-27 15:46 EDT-0400
If you would be happy with a cheap case. The DIYPC case has decent cooling and room. You could get a GTX 1060 6GB and case for about $300 going this route. This case and card will likely be fairly loud.
PCPartPicker part list /
Price breakdown by merchant
Video Card: EVGA - GeForce GTX 1060 6GB 6GB GAMING Video Card ($269.98 @ Newegg)
Case: DIYPC - Alnitak-BK ATX Mid Tower Case ($37.98 @ Newegg)
Total: $307.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-27 15:49 EDT-0400
As prices are starting to come back down to Earth. You may want to wait and see if these come back down to their April levels. In which case you may be able to get the nice NZXT case and GTX 1060 6GB.