Sadly, AMD is really behind in the CPU market right now. Even something as impressive-sounding as eight cores gets clobbered by an i3 in gaming. You'll want to go Intel for sure. Also, most CPUs come with their own fan and thermal paste, so you can save a little there.
Do you need absolutely need the PCI-E drive? It cuts into the budget really hard, especially since you also need a full tower - those start at ~$75 and quickly ramp up! I think a regular SSD will be plenty fast for your needs. I assume you'll also want an HDD since 240GB isn't very much, and you mentioned wanting more storage. Let me know if you really want the PCI-E drive, but this is what I got:
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | Intel Core i3-6100 3.7GHz Dual-Core Processor | $109.99 |
| Motherboard | MSI H170A PC Mate ATX LGA1151 Motherboard | $64.99 |
| Memory | Team Elite Plus 16GB (2 x 8GB) DDR4-2133 Memory | $51.99 @ Newegg |
| Storage | Mushkin ECO3 240GB 2.5" Solid State Drive | $48.99 @ Amazon |
| Storage | Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive | $47.49 @ OutletPC |
| Case | NZXT Source 530 ATX Full Tower Case | $82.99 @ SuperBiiz |
| Power Supply | EVGA 750W 80+ Bronze Certified Semi-Modular ATX Power Supply | $48.99 @ NCIX US |
| Keyboard | AmazonBasics KU-0833 +MSU0939 Wired Standard Keyboard w/Optical Mouse | $14.95 @ Amazon |
| Other | Radeon RX 480 8GB | $229.99 |
| Prices include shipping, taxes, rebates, and discounts | ||
| Total (before mail-in rebates) | $730.37 | |
| Mail-in rebates | -$30.00 | |
| Total | $700.37 | |
| Generated by PCPartPicker 2016-06-23 02:02 EDT-0400 |
The i3 and mobo are a combo deal from Micro Center, and there's another $10 mail in rebate on the mobo. You could get by with a much cheaper, smaller mobo, but it'd look goofy in a full tower.
The RX 480 will come out in about a week. With that in your setup, you'll be maxing out the majority of games at 1080p60. If this costs too much, you could go with a 4GB RX 480. If you need Windows in the budget (I wasn't quite sure based on what you said), you could drop down to an RX 470 or 460 - they're not out yet either, but they're going to beat whatever's currently in their price range for sure. Sadly, we don't know when those two are being released, only the 480.