Final Notice, Please Grade This Build

I would change the cpu to a 4570k, putz a z87 mobo (asus gene/hero), 1600Mhz memory (gskill), samsung ssd (get away from that kingston, I have one and it gets demolished by the samsung 840). Other than that I also have a 770 but since you are a casual gamer you can downgrade it to a 760 if the budget doesn't allow.

I'd agree about changing the ram and SSD.
The HCG620 is a good PSU, but the very similar Seasonic M12 II 620 is cheaper (though shorter warranty). Also worth a look are the Antec TP-650C offers you 80Plus Gold efficiency but no modular cables ($60) or the EVAG Nex650G which gives you both for $80.

Re: 3570K vs 4670K. You can make arguments either way. Neither is going to underperform really. Z87 gives a few newer features and upgrade path potential, Z77 is likely to be a bit cheaper. With a Hyper 212 Evo you aren't going to be going for record breaking overclocks, so I wouldn't have concerns about the 3570k being a better overclocker.
I disagree with the poster above about motherboard spending. I reckon $150 is plenty for Z77orZ87, the Asus Gene/Hero are very nice but don't really offer good value for money. Gigabyte, Asus and ASRock all have very commendable boards for both chipsets in that bracket.

I'm not sure a 4Gb GTX770 is a good investment. It's up to $80 more than a 2Gb GTX770, and in the vast majority of applications, no better. If Vram is a genuine concern (high resolutions, more demanding games on the horizon) then an R9 280X or R9 290 might be a better choice.

There's nothing wrong with the HCG620 or M12 II 620. Both are made by Seasonic, both are 80Plus Bronze rated, and both have similar configurations. If they were a bit cheaper, they'd be a conclusive winner.
The TP-650C I mentioned above is also made by Seasonic, 80Plus Gold rated, Haswell certified, and has a 5 year warranty. For $60 its an absolute steal.
If modular cables are important, the EVGA model I mentioned is probably better value than either of the first two.

In terms of cost/performance, the 3570K is almost undoubtedly better. Its cheaper to buy, and the motherboards to fit it are generally a bit cheaper too. That's around $40-50 saved. The 4670K on the other hand, uses a newer CPU socket that will see future upgrades, and numerous minor advancements in tech. It's not the value winner, but if covering your bases for the future is important, it's perhaps the one to go for. If your needs change, the ability to fit a next generation i7 has inherent value.

Given the various issues with the R9 290 and 290X I wouldn't be in a massive rush to jump on that bandwagon yet. Once non-reference cards start arriving the 290 will probably be worth serious consideration. A few applications can currently use more than 2Gb of Vram, but they are fairly niche (the most widely cited examples are BF4 and Skyrim mods) and limited to very high texture qualities or resolutions. In the case of BF4, the GTX770 still performs absolutely fine, it doesn't get bottlenecked by only having 2Gb of vram, even when running at 2560*1440.

"Future proofing" in general is a fools errand. Spending extra money on the logic you might save some at an indeterminate point, if something else you can't predict might happen, is basically gambling. I've seen people on here suggest buying PSUs dramatically larger than needed on this logic, despite the fact that CPUs and (to a degree) GPUs are all becoming more efficient, or to cover a future multiple GPU setup which very few people need.

If I was buying today, I'd have no issues with getting a 2GB GTX770.

I aim to be ridiculous.

$1000 is a decent amount to spend. If you want a GTX770 it's probably going to mean a concession somewhere else to keep it under budget, I ditched the SSD.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-3570K 3.4GHz Quad-Core Processor ($199.99 @ Amazon)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: ASRock Z77 Extreme4 ATX LGA1155 Motherboard ($124.98 @ OutletPC)
Memory: Team Vulcan 8GB (2 x 4GB) DDR3-1600 Memory ($53.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Video Card: Gigabyte GeForce GTX 770 2GB Video Card ($329.99 @ NCIX US)
Case: NZXT Source 210 Elite (Black) ATX Mid Tower Case ($42.83 @ Amazon)
Power Supply: Antec TruePower Classic 650W 80+ Gold Certified Semi-Modular ATX Power Supply ($59.99 @ Newegg)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($14.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($89.00 @ Amazon)
Total: $1005.72
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-11-19 06:43 EST-0500)

I aim to be ridiculous.

$1000 is a decent amount to spend. If you want a GTX770 it's probably going to mean a concession somewhere else to keep it under budget, I ditched the SSD.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i5-3570K 3.4GHz Quad-Core Processor ($199.99 @ Amazon)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: ASRock Z77 Extreme4 ATX LGA1155 Motherboard ($124.98 @ OutletPC)
Memory: Team Vulcan 8GB (2 x 4GB) DDR3-1600 Memory ($53.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Video Card: Gigabyte GeForce GTX 770 2GB Video Card ($329.99 @ NCIX US)
Case: NZXT Source 210 Elite (Black) ATX Mid Tower Case ($42.83 @ Amazon)
Power Supply: Antec TruePower Classic 650W 80+ Gold Certified Semi-Modular ATX Power Supply ($59.99 @ Newegg)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($14.99 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($89.00 @ Amazon)
Total: $1005.72
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-11-19 06:43 EST-0500)

If you decide you don't want to overclock, then you can downgrade both the processor, the motherboard and the cpu cooler.
An entry level i5, probably the 3470/3350P or 4430/4440/4570 is a better value solution as they are around $30-40 cheaper than the k-series variants, and out of the box are very similar in performance.
These can be paired with either a cheap B75/B85 chipset for the best bang/buck OR a H77/H87 board for a better quality solution. This could save at least another $50-70.
As for coolers, the Intel stock cooler works fine, but in my mind is a little whiney as I could hear mine quite clearly above the 120mm and 200mm case fans I have. There's no real incentive to replace it though.

Every computer component (especially graphics cards) are subject to diminishing returns. The GTX770 is significantly better than a GTX760, but it is also a lot more expensive. Whether or not the cost is truly justifiable is up to you. An $800 build is going to play the same games perfectly well, so the extra you spend is really extending the period you are likely to be able to play at maximum settings. There isn't really a sweet spot in terms of performance, but I'd imagine that the best value is somewhere around the HD7870/GTX660 level, as they can play all the current games at 1080P without any real issues.