You can add an SSD later. This should be plenty for your needs. The ROG Strix B350 is way too expensive and not needed for your budget. 1080p is easy for the 1060 6GB.
PCPartPicker part list:
https://pcpartpicker.com/list/sFRY6X
Price breakdown by merchant: https://pcpartpicker.com/list/sFRY6X/by_merchant/
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor ($198.89 @ OutletPC)
Motherboard: MSI - B350 PC MATE ATX AM4 Motherboard ($65.98 @ Newegg)
Memory: Crucial - 8GB (2 x 4GB) DDR4-2400 Memory ($87.99 @ SuperBiiz)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($46.68 @ OutletPC)
Video Card: Gigabyte - GeForce GTX 1060 6GB 6GB G1 Gaming Video Card ($336.17 @ PCM)
Case: Corsair - 200R ATX Mid Tower Case ($39.99 @ Newegg)
Power Supply: Corsair - CX (2017) 450W 80+ Bronze Certified ATX Power Supply ($48.98 @ Newegg)
Total: $824.68
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-01-11 06:56 EST-0500
$4 over. What country are you in? You can potentially save money if you order there.