Wait, you're gaming on a 4K monitor with an i5-760 & RX 480 at the moment? That's definitely going to be pretty poor performance all-round. The sluggish aspect in browsing may well just require a new OS install (and an SSD if possible).
Honestly, 4K is a tough ask - depending on what you're looking for specifically. 60FPS on what settings? What games do you intend to play? You're highly unlikely to have to worry about 60+FPS for a while yet - you need some of the highest end components (consumer grade, at least) to achieve 60 consistently on max/ultra.
The i7-6700K has shown decent gains over the i5-6600K at 4K, and should last you longer at 4K overall.
The RX480 is not going to be ideal either. With a worthwhile CPU (6600K, 6700K etc), you'll probably be able to achieve playable ~30FPS on reasonable settings, but 60FPS is going to be rare at 4K (without using low presets etc)
At the moment, the GTX 1070 is probably the minimum for decent 4K gaming, with the GTX 1080 or Titan X(P) being preferred. A Fury X may do it, and might be a little more beneficial given your existing monitor......but it is costly.
If your PSU is also 7+ years old, I would replace that before investing heavily in new components.
To get the 'ideal', you'd likely be a little over budget. Using the German PCPartPicker, it looks to be around 1,350 EUR
Shopping in your own country may well make it a little more affordable but, depending on your location, it may also increase prices.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i7-6700K 4.0GHz Quad-Core Processor (€332.94 @ Mindfactory)
Motherboard: ASRock Z170A-X1/3.1 ATX LGA1151 Motherboard (€99.05 @ Amazon Deutschland)
Memory: Crucial Ballistix Sport LT 16GB (2 x 8GB) DDR4-2400 Memory (€73.18 @ Amazon Deutschland)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive (€97.84 @ Mindfactory)
Video Card: EVGA GeForce GTX 1080 8GB Superclocked Gaming ACX 3.0 Video Card (€658.45 @ Mindfactory)
Power Supply: EVGA SuperNOVA GS 550W 80+ Gold Certified Fully-Modular ATX Power Supply (€93.54 @ Mindfactory)
Total: €1355.00
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-09-19 21:41 CEST+0200
This does factor in and SSD & new PSU, but would mean reusing your Case, Cooler & HDD storage
You could always drop that for now, and be right around your budget.
If you wanted the SSD & PSU, you can either drop the CPU to a 6600K and pair with the 1080:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i5-6600K 3.5GHz Quad-Core Processor (€238.21 @ Amazon Deutschland)
Motherboard: ASRock Z170A-X1/3.1 ATX LGA1151 Motherboard (€99.05 @ Amazon Deutschland)
Memory: Crucial Ballistix Sport LT 16GB (2 x 8GB) DDR4-2400 Memory (€73.18 @ Amazon Deutschland)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive (€97.84 @ Mindfactory)
Video Card: EVGA GeForce GTX 1080 8GB Superclocked Gaming ACX 3.0 Video Card (€658.45 @ Mindfactory)
Power Supply: EVGA SuperNOVA GS 550W 80+ Gold Certified Fully-Modular ATX Power Supply (€93.54 @ Mindfactory)
Total: €1260.27
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-09-19 21:45 CEST+0200
Or, drop the 1080 and pair a 1070 with a 6700K:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Core i7-6700K 4.0GHz Quad-Core Processor (€332.94 @ Mindfactory)
Motherboard: ASRock Z170A-X1/3.1 ATX LGA1151 Motherboard (€99.05 @ Amazon Deutschland)
Memory: Crucial Ballistix Sport LT 16GB (2 x 8GB) DDR4-2400 Memory (€73.18 @ Amazon Deutschland)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive (€97.84 @ Mindfactory)
Video Card: Gigabyte GeForce GTX 1070 8GB Windforce OC Video Card (€442.11 @ Mindfactory)
Power Supply: EVGA SuperNOVA GS 550W 80+ Gold Certified Fully-Modular ATX Power Supply (€93.54 @ Mindfactory)
Total: €1138.66
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-09-19 21:44 CEST+0200