Reach level 10 with one of your characters. If you're starting with a new character, you should unlock this fairly early in your first playthrough.

If you already have this achievement on the Xbox 360, then you can unlock this achievement by transferring over your level 10+ character, and then gaining another level. This will unlock both the Reach level 5 and Reach level 10 achievements. 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.