I installed everything as MisterBlack said, and the intro movies load fine, but the actual in-game cinematics are just black screens. What do I do? I have a 2048MB ATI AMD Radeon R9 200 Series and a AMD FX-6300 if any of that matters (thanks speccy, easy to look up now). I've experienced severe issues in the past with the grass in this game, but the movies would load fine.
I downloaded a Dantooine saved game since I haven't gotten that far yet so I could see some actual grass in the game and I can confirm your issue. I have pretty much an identical setup to what you're running: AMD-FX 6300, 8 GB of ram and a AMD Radeon R9 270X all running on Windows 8.1 Pro (64-bit). After some research through Google, it seems to be a common problem with ATI/AMD cards. (I actually seem to recall having similar problems with a nVidia card in the past so it may not be just AMD cards, can't confirm that since I don't have a system with an nVidia card up and running on it to test.)
It seems the only solution for now is to disable grass in the graphics options. It sure doesn't look as pretty, but at least you won't have to look at graphics corruption/glitches anymore. Hopefully in the future either AMD fixes this issue or GOG/The Community come up with a patch or workaround.
As for your black screen on in-game cinematics, I'm not having that issue. Even the ones triggered in game beyond the intro (not to be mistaken for the in-game rendered cutscenes), play without an issue.
All I can say really are the stereotypical answers to these types of problems since I'm not entirely sure what's happening on your end, could be a mistake of your own (It happens), or it could be an issue that I have no idea how to solve that is somehow unique.
I'm currently running the AMD Catalyst 14.9.2 beta drivers, so I'd just make sure you got the latest 14.9 drivers off the AMD site if you haven't already. (Not the betas, the stable ones should be good enough.)
Also check that you removed the lines showed in step 7 in the [Star Wars: KOTOR (1024x768 interface)] section inside the patches.ini file. There will be duplicate entries of those lines for each resolution interface so it must be very specific.
If you don't want to do that yourself, or you if you want to be sure, I did upload my edited patches.ini to Dropbox: https://www.dropbox.com/s/hd2earq3vxztcx6/patches.ini?dl=0
Just replace the one that came with UniWS.zip, restore your original EXE, and try patching again.
I hope that helps. It's about all I have at the moment.