Baseline of your computer should consist of:
16GB of
this RAM - $90
i7 3770K - $330
AMD videocard 7870/7950/7970 (they work way better in Vegas for rendering) - $250-400
CoolerMaster Hyper 212 (if the case allows for it) or CoolerMaster Gemini II (lower profile)
ASRock Z77 Pro4-M - $110
Seasonic M12II 620W PSU - $90
Optical drive of your choice
The case is a tricky choice. If you go down to an ITX form factor, you get a lot more options in terms of cases as it's a really small motherboard. However, the price goes up to $150 for the motherboard. HTPC cases (the horizontal ones) are generally more expensive than standard cases. What you can do is just get a standard case and put it on its side. If that doesn't sound appealing, then you're looking at $100-150 for a case. If you allow for a bit more wiggle room on case design, there's a number of options out there. Take a gander at:
Silverstone PS07, TJ08, Lian Li PC-A04, PC-V351, Fractal Arc Mini, Define Mini. If this isn't happening for a few months, then maybe
this will be out, which should come in around $100.
The reason why I put the 3770K in there is that it is a lot better at rendering video than the 3570K. If shaving off time here and there for rendering isn't worth $100 for you, then you can drop it down to the 3570K.
HDDs is the other touchy part, as prices are still stupidly high. Do you currently have a storage system on your network, or will you be looking to use this PC to store most of your videos?