Looks like you will have room for the radiator, NZXT list top and front mounts for 280mm rads.
https://www.nzxt.com/products/phantom-530
As far as the build for this money I would be adding an SSD for boot and program files, makes thing load that much faster. I only added a budget 120GB SSD and downsized your HDD from 3TB to 2TB to keep the cost relatively the same. I would get a 240GB SSD though.
PCPartPicker part list:
https://pcpartpicker.com/list/DTxRCy
Price breakdown by merchant: https://pcpartpicker.com/list/DTxRCy/by_merchant/
CPU: Intel Core i5-7600K 3.8GHz Quad-Core Processor ($239.99 @ SuperBiiz)
CPU Cooler: Corsair H110i 113.0 CFM Liquid CPU Cooler ($109.99 @ Newegg)
Motherboard: Gigabyte GA-Z270XP-SLI ATX LGA1151 Motherboard ($134.99 @ SuperBiiz)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory ($106.88 @ OutletPC)
Storage: ADATA Premier SP550 120GB 2.5" Solid State Drive ($48.89 @ OutletPC)
Storage: Seagate Barracuda 2TB 3.5" 7200RPM Internal Hard Drive ($64.99 @ SuperBiiz)
Video Card: Gigabyte GeForce GTX 1070 8GB G1 Gaming Video Card ($399.99 @ Newegg)
Case: NZXT Phantom 530 (Black) ATX Full Tower Case ($108.99 @ NCIX US)
Power Supply: SeaSonic G 550W 80+ Gold Certified Semi-Modular ATX Power Supply ($59.99 @ SuperBiiz)
Monitor: Acer G257HU smidpx 25.0" 2560x1440 60Hz Monitor ($254.99 @ B&H)
Other: Windows 10 ($119.00)
Total: $1648.69
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-01-23 09:07 EST-0500