The description of this Trophy might make it seem hard but it is actually very doable if you consider a few things.
First of all I would suggest doing two playthroughs. Do the first one on easy so you can enjoy the story and get all the collectables and misc Trophies out of the way. I really enjoyed collecting those Audio Diaries since they actually contributed to the story much unlike the stuff in other games.
Once you got all that out of the way it's time for action. Start a new game and select Survivor difficulty.
First thing you wanna do is hit

as soon as you can and go into the options. Scroll down and make sure to...
Disable Vita Chambers - ON <-- the important part
You should cling to the following advice throughout the whole playthrough:
1. Save, save, save - after every fight, during big fights, before hacking things. There is no such thing as saving too much.
2. Hack everything - especially the First Aid stations and the security devices. That will make your life a lot easier. Try to get your hands on some Automatic Hack Tools.
3. The Research Camera is your friend - max the research data for every enemy. You'll need it at the end.
4. Focus on upgrading a few weapons - the ones that worked best for me were the Shotgun, Grenade Launcher and Crossbow. Save the Launcher ammo for Big Daddies. I neglected the Crossbow for a long time...how stupid I was. Crossbow bolt to the head is an insta kill on every Splicer and when upgraded you can pick up used bolts on killed enemies.
5. Use your wrench - I think I beat half the game only with the wrench. Get all the tonics that affect it (Sport Boost, Wrench Jockey, Bloodlust). This will actually be the weapon you beat the boss with but more of that later
6. Don't waste Adam on minor Plasmids - focus on the important ones (Electro Bolt, Incinerate!, Security Bullseye, Hypnotize Big Daddy, Telekinesis, Winter Blast). They are everything you need really.
7. Big Daddy fights - the first few will be a pain. Make sure to use Security devices if available. They will get considerably easier later on. As soon as you have Security Bullseye...make it your best friend. It will help you to take out Big Daddies without putting your life on the line. After you have dealt with the Little Sisters in a level always use Hypnotize Big Daddy to get yourself a very handy bodyguard.
8. Little Sisters - always Rescue them. It will give you more Adam in the end.
9. Camouflage - get the Chameleon tonic. With this you'll turn invisible while standing still. Especially handy when you have to hold out against waves of enemies.
*** Spoiler - click to reveal ***10. Boss Fight - the boss was a joke with the following method...
Here's what you need:
- Wrench
- Plasmids (Security Bullseye)
- Combat Tonics (Armored Shell 1+2, Static Discharge 1+2, Wrench Jockey 1+2)
- Engineering Tonics are not important here
- Physical Tonics (Bloodlust, Sport Boost 1+2, Medical Expert 1+2+3)
As soon as you are in the fight get close to the boss. Circle him and hit him with the Wrench. He will rotate constantly but doesn't hit you. Bloodlust will give you a small amount of health from every blow you land so in case he hits you with one of his elemental attacks...don't worry. Should you run low on health just use a med kit. Medical Expert will make sure that you get the most out of it. Starting in round 2 the boss will raise a security alarm. Don't panic though...that is what you have Security Bullseye for. The bots will help you instead now. Keep circling and wrenching him at all times. In the 3rd and last round he will also call some Splicers but since you constantly get health from using the Wrench they are nothing to worry about. Using this method the fight shouldn't take longer than 4-5 minutes.
I also found a video showing this exact method...
Video credits to AdamSpencer87 on Youtube.
Since the difficulty Trophies are stackable you'll also get...