To give you a reference point, I have an quad core @ 3.3GHz that I built in 2012 for gaming. Renders 1080p60 video just fine. Only hits about 75 degrees C under heavy load. I have an SSD / HDD combo, and it is really nice, but not essential. I don't know what the prices for parts would be now, but certainly a lot less than what I bought them for 5 years ago.
My specs:
CPU: Intel i5 2500K
Mobo: Asus P8Z68
Gfx: Radeon HD 6850 Gigabyte brand
RAM: 2x 8GB DDR3 Corsair Vengeance
SSD: 128GB Kingston V300
HDD: 2x 4TB Seagate ST4000
Power Supply: Seasonic X-650
Case: CoolerMaster HAF 922
For graphics card, just buy the best one you can afford, as these get outdated so quickly. On-board RAM on the graphics card will help a bit for video rendering, as well. You should be able to get by fine with 8GB of RAM, but if you can swing it, 16 is nice.
Nice thing about the SSD / HDD though is that you don't need to buy a huge SSD, you can buy 250GB for about $100 (USD) these days, and then get like a 4TB HDD to store all your videos on. One thing I will say is careful on the SSD manufacturer. I have had one die on me before a year, and one that was dead on arrival before I even installed it. Two different manufacturers, both very reputable. So check reviews before buying if you go that route. I would say if you do HDD though to get a faster processor, but if you were on a budget, you could go as slow as I have with an SSD and still be all right.
Make sure you get a good power supply that can support all the components you have!
On a budget of $200... honestly, I'm not sure that will do it. You could maybe try computer shops that sell used parts? What are the rest of your specs?