Your choice of CPU (Intel) is really not limited here. Since you don't need any fancy new instruction sets, lots of cores, or even hyperthreading there are lots to choose from. Since you are going from a laptop to a brand new desktop CPU, your performance is going to be vastly better.
You have not laid out a budget yet, so that is why it is somewhat hard to recommend. But from what I read (even from you posted) a K series (for overclocking) i5 should be fine. Heck, even an i3 should work (although i3's aren't overclockable).
When you set a budget, don't forget you will need a display, keyboard, and mouse!
The difference between the i5 and i7 (aside from hyperthreading) is .1 GHz on both base and turbo frequencies (<3% difference). The price difference is about $50 right now as the 4770k is currently discounted. As stated earlier, you could get an even better deal at MicroCenter. A 3% difference at 30 FPS would net you almost 1 more FPS (theoretically). Only you can decide if that is worth it to you
For overclocking you need to ensure that your case is big enough for the cooler you choose and that you ensure you have enough fans and that they are ventilating the case properly. As stated, even with a good motherboard, well-cooled case, and great cooler the CPU may not overclock well. But you won't know that until you try it.
You could buy the i5-4670k, the Asus Z87M-Plus, a Noctua NH-D14 (or a Coolermaster Neptune 280L, if you want to try a closed loop cooler) and a CoolerMaster HAF 932. Then see how your stock performance is. If you want more speed, start by allowing the system to choose the overclock. If you need even more, you can start tweaking (carefully!) to see if you can bump it up. I think, though, that with a modern processor (and modest - probably the automatic) overclock you will be pleasantly surprised with a system that will last you more then a few years. You are probably talking roughly ~900 for this build, and that is before the memory, power supply, and graphics card is added in. So you are probably looking at ~$1500 for a build like this (depending on how much you are going to spend on a monitor - which drives the decision of graphics card you are going to want to buy). It could push up to $2000.....is that in your range? You could easily lower this by choosing a slower/non-overclocking/i3......and only lose a little performance (probably between 10-25% depending on where you went) but the cost would drop as well....
But, as noted above, using an i3 would still be a great choice (since your only current parameter is single threaded performance) but with a much lower cost. You no longer need a big base with lots of fans and room for a big cooler/radiator. The price for an i3 build then probably drops to about ~$500 for the base components (processor, motherboard, case, ssd) vs ~$900 above with a ~1% drop in single threaded performance out of the box. Is 1% (and the chance to overclock to add some more) worth $400 to you? Only you can answer that question.... This system is still quite powerful. And, if you buy it now and decide you want an i5 or i7, you could buy it and replace your CPU (and add an aftermarket heat sink).
So it really depends on if you are after squeezing out every ounce of performance you can or if you are budget constrained (aren't we all?) and what components will best fit those needs. But, as has already been stated, going from a VAIO to a brand new desktop system you should be pleasantly surprised (even with an i3).