I got this far:
SELECT @rownum:=@rownum+1 'bd_rank', u.bd_id, u.bd_likes FROM bd_user u, (SELECT @rownum:=0) r ORDER BY bd_likes DESC
Now I just need to figure out how to:
UPDATE videos SET videos.bd_rank=results.(SELECT same above) WHERE videos.bd_id=results.bd_id
...or something like that. I'm not good with compound statements.