Looking to upgrade current PC

[WhiteSnow]

Current PC
Mobo = Gigabyte Z87X-D3H
CPU = Intel i5-4670K
Socket = LGA1150
GPU = Sli GTX 770 (Gigabyte Windforce)
PSU = Seasonic Snow Silent 1050 Watt - 80+ Platinum
RAM = Crucial Ballistix Sport 16 GB DDR3 - 1600 MHz

Approximate Purchase Date: Variable, I can wait a couple months if thats best.
Budget Range: $2200 - 2000 Euros
System Usage from Most to Least Important: Multitasking of the following things: Gaming, Movies, Browsing, Streaming (Just started to experiment with streaming)
Are you buying a monitor: No
Parts to Upgrade: CPU, MOBO, RAM, GPU
Do you need to buy OS: No
Preferred Website(s) for Parts: No preference
Location: Netherlands
Parts Preferences: Intel CPU / Nvidia GPU. (No Gigabyte)
Overclocking: Maybe
SLI or Crossfire: Maybe
Your Monitor Resolution: Currently1920x1080 But if 4k is interesting enough I might want the option to upgrade.

Comments
I am looking to upgrade as I am experiencing some lag or not enough power for the job. Multitasking is VERY important as I usually play games and watch streams at the same time.

Also modded minecraft and streaming show that my CPU is maxed out or close to max.
Games I play: CS-GO, Minecraft, Fallout 4, Soon Battlefield 1, basically a lot of different games. Also some yet to come.

I am not looking for the most silent parts, however I don't like an airplane in my room. Performance is important.

Currently selected parts:
EVGA GTX 1080 FTW Gaming

Intel i7-4790k

or the yet to come 7th generation intel i7.

 

[Icaraeus]

Most I would change is CPU from Intel i5 to i7 from the same socket. I wouldn't upgrade anything else. If you want to go for 4K though, then GTX 1080 SLI is a good choice. Would cost you US$1500 at most.

 

[WhiteSnow]

Yes I have been looking at the same socket as well, BUT i cant help but feel bad for not getting something more powerfull. I am just afraid that the i7 version is gonna be maxed out fast as well. Remember I max out on minecraft / sometimes CS GO when also watching streams / streaming allready. So will the newer i7 7th generation (rumored coming late this year) help me more?

And what about cores? I dont know much about that. Will it help me significantly more if I get say 6 core CPU instead of the 4 from the i5 and i7 (1150 socket) ?

Edit: I always have multiple Chrome tabs open when playing a game. (1 stream, and browsing the web) like at this moment I have TomsHardware open, couple google tabs and a Twitch Stream (high Quality) AND I am playing minecraft.

 

[Icaraeus]

Intel i7 won't have any problems with streaming and playing games at the same time. You can use existing socket. No need for motherboard switch. For streaming and gaming, 4 cores + HT is all you need. You don't need extreme Intel i7s. 16GiB RAM is likely all you need, but I'm not entirely certain as I never will up 8GiB RAM anyway.