Well afaik there would be 4 ways of doing this:
.DM2 demo file - I think jitspoe and S8N did one of these that played in the background, complete with music, for the q2 mod version.
Pros: Small file size, renders in game
Cons: Hard to edit, no 'video effects' or transitions possible.
.CIN cinematic file - I tried converting a youtube video to one of these earlier with a variety of ancient tools, only got the sound to work, screen was black.
Pros: Plays back on virtually any computer, can use video effects
Cons: Difficult to create, low quality, large file size
RScripts - These have been used for animated menu backgrounds
Pros: Easy to edit for different configurations
Cons: Eats up tons of memory, takes ages to load, CPU heavy, no sound
Add support for another video format (RoQ)
Pros: Reasonable quality, general compatibility, can use video effects
Cons: Relatively large file size, needs to be ported or adapted from KMQuake2 or ioQ3
So at the moment, it's not too practical to do this.