Will an Intel i7 4790K CPU work with and AMD r9 380 GPU?

yes

R9 380 is a good choice; go for it.

You can find some models of 390 or 970 in that price range.
If you have to keep the i7 it will still perform well, personally I would take an i5/Xeon with a stronger graphics card.

Is this strictly for gaming, or will you be doing rendering or similar?

You could get these, and spend the extra money elsewhere (Better cooling, better motherboard, larger SSD etc)

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4690K 3.5GHz Quad-Core Processor (£161.94 @ Aria PC)
Video Card: MSI GeForce GTX 970 4GB Twin Frozr V Video Card (£259.00 @ Aria PC)
Total: £420.94
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-07-27 19:01 BST+0100

Easily an upgrade, it is far superior.
The 4690k wont bottleneck any GPU out there.

Re: GTA 6

RECOMMENDED REQUIREMENTS (1080p)
Intel CPU: Core i5-3470
AMD CPU: FX-8350
Nvidia GPU: GeForce GTX 660
AMD GPU: Radeon HD 7870
RAM: 8GB
OS: Windows 8 64-bit
Direct X: Dx 11
HDD Space: 65GB

MINIMUM REQUIREMENTS (1080p)
Intel CPU: Core 2 Quad Q6600
AMD CPU: Phenom 9850 Quad-Core Black Edition
Nvidia GPU: GeForce 9800 GT
AMD GPU: Radeon HD 4870
RAM: 4GB
OS: Windows 7 64-bit
Direct X: Dx 10
HDD Space: 65GB

For me, and I like being both clever and thrifty, I would go with a dual core Pentium G3258 clocked at 4 - 4.4 Ghz using a HSF like the Cooler Master 212 or similar. There are many fine HSF out there (Xigmatek are my favorite ones) both new or used. You will game at about 50 deg. C on the CPU. The mobo should be a Z97 chipset to get the best stable OC. Nice mobos with the Z97 can be had from eBay. I got a positively great deal on a very slightly used one (Gigabyte Z97X-UD3H) for about 60 quid. The G3258 is unlocked and performs within a couple FPS of the I7-4670K in most games. Hey just look at reviews on the G3258 on youtube and other sites. I have one and run it at 4.4 and I have a i5 4690K as well. They really feel about the same. I have a Radeon 270x (predecessor to the R7 370) with that G3258 and run all kinds of games capably on it. (War Thunder, WoW, SWTOR etc. Can you say 100 FPS on high settings?

Why spend money for not much return? Save a couple hundred quid on that decision alone. For a GPU, I'd go with the R7 370 2GB. I saw a nice MSI one with 4GB on Amazon UK for L140.00. The cpu/gpu combo I just put together here goes for just under 200 quid. It'll run the game just fine and look great. This computer will be rally quick. Now, if you want to do animation, video editing or other things that are multi-thread intensive, this might be what you want. But - if you want a fun and snappy machine for general use and some fun gaming, this one will fill the bill and you'll save yourself 250 pounds.

I'll tell you another secret: You don't need 8 GB ram. You are fine with 4 GB is you use 32-Bit Windows. That's right. The games are 32 bit anyhow and your machine will seem just as fast with the 32 bit OS. Save another 40 quid on the ram that way. You won't need a big PSU either. Now you'll have money left over for a big monitor and nice other things. Now you have at least 300 quid to use elsewhere.

Those two components definitely should be compatible!