Reach level 50 with one of your characters. If you're starting with a fresh character, you most likely will not unlock this until your second playthrough.

If you already have this achievement on the Xbox 360, then you can unlock this achievement by transferring over your level 50+ character, and then gaining another level. This will unlock all of the level up achievements (Reach level 5, 10, 25, and 50). I had transferred over my level 56 Axton, and when I reached level 57, all 4 achievements popped, so I can confirm that this definitely works.

Apparently the achievement will also pop if you join a friend's lobby with a character that's the required level for the achievement. I can't confirm it myself, but ELBaRTii in the comments below said that's how it unlocked for him. If leveling up your character one more time is too time consuming/difficult, then this method should make things much easier.