DXTory

Step 1: http://i.imgur.com/JemUYBJ.png
Open up DXTory. Choose Lagarith as your codec (lossless video encoding) and make sure File Output isn't ticked, but that DirectShow Output is. The screenshot also shows the Lagarith settings window (all of which is default iirc). It's possible to use other codecs, but Lagarith isn't hardware intensive and doesn't diminish the video quality.

Step 2: http://i.imgur.com/xFq8ls6.png
Under "Video settings" (the yellow button at the bottom of the DXtory window), make sure it's set to what's shown in the screenshot. If you'd like to stream it at a higher/lower resolution, you can change the values, of course.

Step 3: http://i.imgur.com/cqgwPYl.png
More DXtory settings (under a different tab)

Step 4: http://i.imgur.com/2dvIGOj.png
Make sure that OBS is added under the "ignore list". OBS is listed 3x in the screenshot, because I commonly use 3 different versions of it depending on what I stream.

Step 5: http://i.imgur.com/seBcKXo.png
Make sure you have a hotkey set up for starting the video capture.

Step 6: http://i.imgur.com/5F76uih.png
In OBS, add a new video capture device. Then, select DXtory 1 as your device. Default settings are fine for this page, but feel free to mess around.

Step 7: http://i.imgur.com/vPAc8Z0.jpg & http://i.imgur.com/NTwUh08.jpg
Once you boot up Vice City, press your capture hotkey, and the text in one of your screen's corners will turn from green to orange. It also shows at which framerate you're capturing the game. If it dips below 29 at any point, you're probably using too demanding settings for streaming it and need to scale down a bit.

Step 8: http://i.imgur.com/RaTlA8x.jpg
The game should now show up in OBS when you preview your stream. Ready to go!