This trophy requires you to save every Little Sister, and not kill Grace, Stanley, and Gil.
As you play, you will come across Little Sisters with their Big Daddies. In order to get the Little Sister for yourself, you must kill the Big Daddy. The Little Sister will then start crying at the daddy's corpse, walking up to her will bring the options to pick her up or harvest her. Do NOT harvest her.
Now, you have two options. If you need or want to get the trophy for getting 600 ADAM, you should press square and let the sister lead you to the nearest ADAM filled corpse. The trail is a light pink mist, so follow that and set the sister down at the corpse. Splicers will come and attack so defend yourself and the sister. Little Sisters can drink ADAM twice each before they are full. Once you have let the sister drink ADAM twice, press square to find the nearest vent, and then rescue her. Again, do NOT HARVEST HER. You will get more ADAM, but the trophy will be immediately forfeit if you harvest even one little sister.
Alternatively, you could also take the sister straight to the vent and rescue her without gathering any ADAM. You aren't able to do this when you first find a Little Sister, or in Dionysus Park.
Now, technically, you can't miss any Little Sisters unless you reach Persephone. Before that point, any little sister than you missed from one level will 'migrate' to the next level. Of course, this means that if you decide to leave all non-mandatory sisters for last, you'll have to collect ADAM over and over and over again. Not recommended. I would say that you should pick up the sisters the collect ADAM normally throughout the game. Pressing start will tell you how many little sisters are still left to find in that level. Pressing select will bring up the map. Here you can usually find the exact location of a little sister. Also, keep an eye out for singing. It's the best way to identify the Little Sisters.
Once you have rescued all of the Little Sisters, you should get a trophy for having dealt with all of the Little Sisters. Or if you harvest them, you evil person. Also, be warned. After dealing with all of the Little Sisters in a level, a Big Sister will attack you. You get decent warning, so make sure that your health isn't low and you have enough EVE and ammo to fight her. She's not generally tough, just fast and annoying.
Now, onto the characters. Grace Holloway is found towards the end of Pauper's Drop. You have to battle your way up to her room in the hotel, and use the button behind the suspicious poster of Lamb to open the secret door. There, you will find Grace, who doesn't seem to like you very much. Do not touch her! No gunfire, no melee, no plasmids, nothing! Just take the key, and whatever loot you want and leave! If she dies, the trophy is forfeit. She'll reward you with some loot later on.
Next, Stanley. He is found in Dionysus Park, and has a very fishy demeanor. He wants you to go find the Little Sisters and deal with them... your way. Doing this, Eleanor will tell you the secret to Dionysus Park. After dealing with the sisters (and, subsequently, the Big Sister), you have to make your way back to the train station, where Stanley turns into a big pile of coward. Don't kill him, however much he may deserve it. Take the train key, and the loot inside the room (including an audio diary), then leave.
After, it's Gil. Gil Alexander, to be precise, or Alex the Great, amongst crazy company. Gil is 'found' in Fontaine Futuristics. When you arrive, he will entreat to you to kill him, from an audio diary from the past. Essentially, you have to let him wallow in his crazy state, which is a little backwards, to me, but that's how you get the trophy. In order to NOT kill him, follow his commands about getting the ADAM infused plants, and the data key needed to proceed. Then, when you have taken the data key from the console, DO NOT TOUCH THE CONSOLE AT ALL. Just back away. Leave. Even as Gil asks you to kill him and is very nice about it all, leave. Pressing the button and electrocuting him in the tank does seem the more humane thing, given the circumstances, but it will forfeit the trophy.
Having now saved all of the Little Sisters and the required NPCs, the trophy should pop with a little ding and you should find yourself the proud owner of a silver trophy!
If there's anything I have left out or you don't like, please tell me below and I will amend the guide. Thanks!
Happy trophy hunting.