Video
To get a video cutscene, intro or outro into the game, the video needs to be in a proper format - BINK video format .bik.
Editing the video
The video you recorded and want to use has to be edited. My go-to editor for this is kdenlive. It works very well, it is free and open source, and it supports BINK video as an input, which is great if you want to include subtitles in the video.
My version of kdenlive does not know how to export video straight to .bik so I just export my video to .mp4 and then convert it with RAD Video Tools.
RAD Video Tools
RAD Video Tools is a tool for converting other video formats to BINK .bik that Gothic can use.
Warning
Gothic 1 bink implementation has some problems as you have to set the audio compression to 104 and above in RAD tools to get video to work in Gothic 1.
NicoDE's comment:
Add 100 to the audio compression level when encoding videos, e.g. 104 for level 4 with old sound format (should be mentioned in the RAD Video Tools documentation) for G1 without updated Miles libraries.
Note
Newest Union (1.0m at the time of writing) has a new patch for BINK video playback. The issue with sound should be fixed.