"Play Game" -> "Bullet Soul Caravan Mode"
For this one, you need to improve your leaderboard score in caravan mode with a single character 3 times. Meaning a minimum of 4 plays with that character.
I suppose it is possible to lock yourself out and effectively make this unobtainable if you make your scores too impossible for yourself to beat. So some caution is required.
As per flubberfudge's comments, you can erase your times if you mess this up.
Details: Play Game-> Caravan-> Local Ranking-> Select Reset-> Use bumpers to choose character.
With 3 characters to choose from however, you have plenty of chances to unlock this especially as you go for the other caravan achievements.
It does seem these scores need to be uploaded to xbox live, but correct me if I'm wrong there.