The simplest answer seems to be that if you want to add music to the background of your Vine videos, you need to use an app!
You can add music to Vine and Instagram videos using an app called Background Music For Video by Sepia Software.
It is available from the app store for free!
The steps to using this app are pretty straightforward:- Chose your video from your device’s photo album.
- Chose a track from your iTunes.
- Pick a start time for the music.
- Merge the video and music together.
Then all that's left is to share on the social network you prefer: Instagram or Vine!
The other work around for adding music to your Vines is to record and edit clips beforehand, and then re-upload them to Vine.
This has better effects, but is a little trickier, so rather than explain it I'll just share a useful how to video:
Once you've edited and uploaded your Vines, you will probably want to know how to become "Vine Famous".
Luckily, there is a how to video for that too...