Definitely don't get a low end CX series unit. Also, Thermaltake's low end cases are very cheaply built - a $10-20 increase in case budget can make a huge difference in quality here.
An i3 is a weird choice for a GTX 970. This budget should include i5 in my opinion, even at the cost of the graphics card. It's much easier to just upgrade the graphics card later, and that makes sense too because graphics card performance evolves much faster than CPU performance these days. An i5 in 2 years will cost basically the same as an i5 today, and perform roughly the same too, so why delay the purchase for 2 years? On the other hand, a GTX 970 may seem like beast compared to $200 cards now - but you should be able to get GTX 970 level performance at $200 with NVIDIA Pascal by the end of the year.
I'd much rather spend a little more on the PSU, case and processor to ensure longevity. I'd also buy a 250GB SSD so it doesn't immediately run out of space the moment you install one bigger game on it (say, a modded Fallout 4...); 120GB SSD's also have much higher cost per gigabyte.
i5-6500 $193 after promo code EMCEFGK22
MSI Z170-A PRO $80 with free Steelseries Kinzu v3 mouse (which you can sell 'as new' for $25 if you don't need it)
2x4GB G.Skill NT DDR4-2400 $40
Sapphire R9 380 4GB Nitro $180 AR with free mouse pad (no idea if the mouse pad is good)
Kingston HyperX Fury 240GB $70
Hitachi 1TB $42
EVGA 650 GQ $60 AR
Fractal Design Core 3300 $50
Windows 10 $94 @ amazon
= $809 AR + possible shipping
This is higher quality build, even though strictly speaking gaming performance isn't as good. The GPU should be the only thing you need to upgrade here for the next  ~5 years (and the SSD, PSU and case should last even longer - probably RAM too, since DDR4 is new).
You can overclock the i5-6500 using BCLK (to be fair, the same applies to the i3 above). Should be able to do 4.3-4.4GHz without much fuss, which is a great way to improve CPU performance down the road - all you need to add is a normal 120mm tower cooler like the overrated (though competitively priced) 212 EVO.