Reach level 25 with one of your characters. If you're starting with a new character, you should have no problem getting this achievement on your first playthrough as long as you're doing side quests along the way.

If you already have this achievement on the Xbox 360, then you can unlock this achievement by transferring over your level 25+ character, and then gaining another level. This will unlock the previous level up achievements as well (Reach level 5, 10, and 25). I had transferred over my level 56 Axton, and when I reached level 57, all 4 level up 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 someone mentioned in the comments of the level 50 achievement that this is 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.