I checked and it's quite easy actually. Here's what you do:
- Go to any Pokecenter.
- Go to the Global Terminal (top floor, on the right).
- Choose "Battle Videos" then "View or upload".
- Select "Send Battle Video" and it will get uploaded.
- Make a note of the number. Mine was 74-88678-06337.
Then to view a specific video:
- Go to the Battle Videos section as above.
- Select "View Battle Videos" then "Search by No".
- Put in the number and save it.