To unlock this you must kill a Big Sister without being killed. I doesn't have to be the first one that you encounter, it can be any.
You encounter the Big Sisters when you deal with the final little sister in the area, you can harvest then or save them, it doesn't make a difference.
Before you start the fight with the Big Sister make sure you stock up on full health kits, Eve Hypos and drill fuel. It would also help if you upgraded you drill before going into the fight.
If you have any money left then stock up on ammo for your other weapons. You should also lean the lunge move with your drill.
I just zapped the sister using the Electric plasmid and then lunging at the with the drill. This inflicts massive amounts of damage. You should kill her by the time you run out of fuel and Eve, but just in case you do just shoot her from a distance with whatever you have ammo for.