7 Year Old Computer - Need to upgrade!

Ok, so based on your current PC and your foreseen budget, here's my suggestion for your upgrade:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-6600K 3.5GHz Quad-Core Processor ($239.99 @ Amazon)
CPU Cooler: NZXT Kraken X31 69.5 CFM Liquid CPU Cooler ($73.99 @ Newegg)
Motherboard: Asus Z170-A ATX LGA1151 Motherboard ($159.99 @ SuperBiiz)
Memory: Crucial Ballistix Sport 16GB (2 x 8GB) DDR4-2400 Memory ($112.49 @ Newegg)
Power Supply: EVGA SuperNOVA GS 650W 80+ Gold Certified Fully-Modular ATX Power Supply ($79.99 @ NCIX US)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($15.98 @ Newegg)
Operating System: Microsoft Windows 7 Home Premium SP1 OEM (64-bit) (Purchased For $0.00)
Case Fan: NZXT FZ-120mm LED 59.1 CFM 120mm Fan ($10.99 @ Directron)
Other: NZXT LEXA-NP Black/Silver ATX Mid Tower Computer Case (Purchased)
Other: 1TB Backup HDD (Purchased)
Other: 500GB Solid State Drive (Purchased)
Other: EVGA 01G-P3-1373-AR GeForce GTX 460 (Fermi) Superclocked EE 1GB (Purchased)
Other: D-Link WDA-2320 Rangebooster G Desktop Adapter (Purchased)
Other: Hanns-G HX-191DPB Black 19" 5ms LCD Monitor (Purchased)
Total: $693.42
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-09-11 14:35 EDT-0400

The above parts can be grouped as your "first batch of upgrades" given the $700-$800 initial budget. My reasons for selecting the parts were as follows:

CPU:
It was necessary to upgrade the CPU (with the motherboard and the RAM) first before the GPU to avoid any bottlenecks if it was the other way around. Your preference to upgrade to Skylake was heavily considered. It seemed logical as you wanted to "future-proof", that is, make your build last several years. The recently-released Skylake processors, as of this date, are quite limited. Actually, we only have 2 readily-available processors to choose from - the mainstream Core i5-6600k (3.5GHz) and the performance Core i7-6700k (4.0GHz). With a price difference of $120 between the two, the more affordable i5 has the clear advantage in terms of price-performance over the more expensive (but more powerful) i7. Both processors have 4 cores (but the i7 has Hyper Threading Technology effectively making it perform on 8 threads vs the i5's 4 threads). Both can be overclocked. But the i7 only has a range of 200MHz (from 4.0GHz to 4.2GHz) while the i5 could cover more ground at 400MHz (from 3.5Ghz to 3.9GHz).

MOBO:
Consequently, the with the choice of the recently-released i5-6600k CPU having a Socket 1151, a recently-released and compatible motherboard was chosen - the Asus Z170-A. This motherboard has more than enough features, such as a good number of connectors (SATA, USB, M.2, Thunderbolt, etc.), multi-GPU support for either Nvidia SLI or AMD CrossFire, and 8-Channel Audio with DTS support on ALC892 Audio Codec (you won't be needing that sound card anymore), it will be more than enough to suit your present and future needs. And, not that it matters, but I chose a Black/Silver/Gray color theme to match the other interior components of your rig and your case.

RAM:
Due to the motherboard upgrade, the RAM modules follow. We now have the newer and faster DDR4 memory (again, for "future-proofing"). Although the motherboard can support higher memory speeds (up to 3466MHz on overclock), I opted for the more affordable 2400MHz. I got 2x8GB sticks as I wanted the other 2 slots open, not only for future upgrade (as you probably won't need more than 16GB total), but to ensure that the memory controller is not stressed out during overclocking if all slots are filled. And, again, not that it matters, the same Black/Silver/Gray color theme was selected.

PSU:
Although your current power supply is still working, we cannot discount the possibility that its efficiency (and reliability) may have decreased in the long years it has served its purpose. I recommend to upgrade the PSU right away in this first batch of upgrades for safety purposes (that CPU-motherboard-RAM purchase is not cheap!). The main reason for selecting the EVGA SuperNOVA GS 650W 80+ Gold is for a future GPU upgrade. I was looking along the lines of either a GTX 970, GTX 980, or GTX 980 Ti which requires 500W, 520W, or 600W, respectively. Should you choose any of these GPU's in the future (in the second-batch), this first-batch PSU upgrade will already support it. The EVGA model I chose is a high-quality Tier-1 category power supply that should be good for a long number of years. The 80+ Gold rating also helps in reducing electricity costs in case you'll be running your rig full-time 24/7 (or close to that). It is also fully-modular so you'll only put in the cables you need in case your chassis has poor cable management design (I noticed that the PSU is top-mounted, so, there will be less cables dangling above and interior airflow not be obstructed).

OPTICAL DRIVE:
I chose a basic DVD/CD Writer in this first-batch of upgrades, nothing special here (I'm not sure if you already have one as you didn't mention it in your post).

CPU COOLER:
This shall be required since we have selected an unlocked CPU for overclocking. Replace the stock cooler that comes with the CPU with this aftermarket CPU Cooler. I choose the NZXT (same brand as your case) Kraken X31 69.5 CFM 120mm Liquid CPU Cooler for this purpose so that you can mount the fan/radiator at the 120mm rear fan slot of your current case. I initially wanted to get the more affordable Air Cooler (Cooler Master Hyper 212 Evo) but I cannot find any data regarding your case's dimension/clearances if such air cooler can fit.

CASE FAN:
Just to address your concern about the defective and/or noisy fan/s you currently have. The 120mm fan in this upgrade will be mounted at the front intake. As you already have the rear fan slot to be occupied by the 120mm radiator fan of your CPU Cooler, all that's left is the 80mm fan slot above (exhaust). It's your call if you want to add another fan in that location (this will make your interior airflow to have more negative pressure).

Now, for your "second-batch" of upgrades, it will only comprise of these 2-4 major items:

GPU:
As I mentioned above (in the PSU), the best price-performance upgrade would be any of the following: GTX 970, GTX 980, or GTX 980 Ti. I put the GPU in the second batch because this is one of the more expensive upgrades you will do (and harder to decide). More importantly, your upgraded CPU will not bottleneck your current GTX 460. If you upgrade your GPU first, your current AMD Athlon 64 X2 6000+ will decrease the upgraded GPU's performance.

MONITOR:
This will now be a personal preference depending on which GPU you opt to purchase. All recommended GPU's (GTX 970, GTX 980 and GTX 980 Ti) will definitely max out games played on a 1080p resolution. Going to a 1440p, the GTX 970 will run most games around 50-60-ish fps while the GTX 980 and GTX 980 Ti will have no problem. If your current monitor is still going strong by the time you will do this second-batch of upgrades, you may also opt to go with a dual-monitor setup. This will increase your productivity with 2 screens simultaneously open and partitioned in a way you seem fit (e.g., MS Office/Gaming on the main monitor while Internet/Social Media/Reference in the other monitor).

WI-FI ADAPTER:
By then, you may also get a good wi-fi adapter card with a connectivity that supports the faster 802.11ac. Of course, you may also opt to purchase a new wireless router with the same speed (although not necessary as the 802.11ac has backward compatibility with the slower 802.11a/b/g/n). Though it may seem unnecessary if you have slow ISP package/Internet connection with low Mbps, the added speed of the 802.11ac is beneficial should you be sharing your files with other wireless devices in your local network. In upgrading the Wi-Fi Adapter, it is best to choose a brand/model with antennae stand/extensions for better reception rather than having the antennae fixed at the back of the case.

CASE:
I picked the parts above with the intent of keeping your current case. I share the same comment - that case looks awesome. Wish it had USB 3.0 front headers. You might just have to go with a USB 3.0 Hub connected to the back of your case to get that perk.

So, that's it. Hope all of this info will help you in making a final decision on which parts to actually purchase. Good luck in your upgrades!