You'll get this achievement as part of the story progression.
The Peach Wilkins fight is done without any weapons (except your wrench). Therefore, you'll have to be clever, since other splicers will appear during the fight.
I like to make a sweep of the room, melting any ice in front of the four smaller rooms for easy access. Also during this time, you should pass a security camera. Definitely hack it, so that the enemies will trigger security bots during the fight.
As soon as the mist clears, a metal shield will open up on the north side of the room with a turret behind it. If you can find this area before the mist clears (it's up on a ledge), you should be able to preemptively shock the turret and hack that. There's a nearby metal shield that you can use for cover as well.
After hacking the turret, you can look for goodies in the four smaller rooms. There are plenty of explosive barrels scattered around for you to use. You can also reach the southeastern room from entering the northeastern one, melting the ice in front of the hole in the wall, and crawling through. You will be very well protected in this room.
Anyway, to deal with Peachy himself, I like to just sit behind the aforementioned metal shield near the turret and use Telekinesis to hurl his molotovs back at him. It should take four or five to take him out completely.