No it won't run smoothly. You pretty much reach CPU bottleneck even if overclocked not far past your current GPU in terms of power.
It will be tremendously heavy in terms of bottleneck.
Instead of getting the RX480, get one of these, although in your case I recommend the i3 based on your budget and need for an upgrade in terms of GPU power.
PCPartPicker part list:
http://pcpartpicker.com/list/fhBm99
Price breakdown by merchant: http://pcpartpicker.com/list/fhBm99/by_merchant/
CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor ($198.99 @ SuperBiiz)
Motherboard: Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard ($53.88 @ OutletPC)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($67.99 @ Newegg)
Total: $320.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-08-06 00:41 EDT-0400
i3:
PCPartPicker part list:
http://pcpartpicker.com/list/b2wYxr
Price breakdown by merchant: http://pcpartpicker.com/list/b2wYxr/by_merchant/
CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($110.99 @ SuperBiiz)
Motherboard: Gigabyte GA-H110M-A Micro ATX LGA1151 Motherboard ($53.88 @ OutletPC)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($67.99 @ Newegg)
Total: $232.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-08-06 00:42 EDT-0400