For this achievement you need
- a diamond which is already mined by someone else
- or a Iron, golden or diamond pickaxe, along with diamond ore found somewhere underground.
If you want to mine it simply use one of the 3 above listed pickaxes.
If you already have a diamond in someone elses world or in your own world, simply drop it on the ground and pick it back up. You can use achievement world v3 for this.
Achievement worlds