Maxed Out 
Upgrade all weapons and equipment.
This is impossible on a single playthrough. Try to save as much money as possible on your first game to spend on Power Nodes later. Also, make sure to grab all of them off the walls. After you complete the game, save and reload to continue with your items.
Ragdoll Check 
Force an enemy into a Gravity Panel 5 times.
This can first be done on Chapter 4. The panels on the floor that kill you and everything else are gravity panels. To force enemies through, first use Stasis on them, then walk into them to push them into a gravity well. This doesn't have to be completed on Chapter 4; there are a couple other segments with gravity wells.
Big Spender
Spend 200,000 credits at the store.
You'll get this whilst playing through the game. DON'T SPEND MONEY FOR NO REASON! Trust me, you need it for Power Nodes.
There's Always Peng
Find the Peng Treasure
Peng's Treasure is located directly to your right as you exit the ship on Chapter 1. Unfortunately, its also out of reach. When you come back here in Chapter 11, you will have Kinesis to grab it with. As soon as you step into the landing bay in Chapter 11, it will be to your right, in between the catwalks and over the railing. It is a small golden trophy.
Weapon Kill Trophies
Full Contact
Kill 30 enemies with the Contact Beam
Self-explanatory. Simply buy the weapon, and kill the enemies.
A Cut Above
Kill 30 enemies with the Ripper
Self-explanatory. Simply buy the weapon, and kill the enemies.
Tool Time
Kill 30 enemies with the Plasma Cutter
No your game is not glitched; there just aren't 30 enemies on Chapter 1.
Pusher
Kill 30 enemies with the Force Gun
Self-explanatory. Simply buy the weapon, and kill the enemies.
Live With The Hot Ones
Kill 30 enemies with the Flamethrower
Self-explanatory. Simply buy the weapon, and kill the enemies.
Autofire
Kill 30 enemies with the Pulse Rifle
Self-explanatory. Simply buy the weapon, and kill the enemies.
Eviscerator
Kill 30 enemies with the Line Gun
Self-explanatory. Simply buy the weapon, and kill the enemies.
Secret Trophies
Brute Force
Kill a Brute
You can't miss it. The Brutes are the huge, tank-like necromorphs. You have to fight about four or five in order to complete the game. The first is on Chapter 4. Easiest way to kill one is shoot it with Stasis and shoot it between the shoulders when it's facing you to bring it to it's knees, stasis again go behind and shoot same spot on the back. These things can take some punishment, as long as it's bleeding your hurting it.
Don't get cocky, kid
Survive the ADS Cannon with over 50% shield strength remaining.
This is the segment on Chapter 4 where you have to shoot down the asteroids before they hit the ship and is pretty difficult. Remember you have two guns, and you can hold down triggers, but the guns will overheat. Try to only concentrate on the ones headed straight for you and the huge ones that split. I noticed that some of the asteroids will fly off screen and won't damage you. This takes some practice and might take a few tries.
Playing Catch
Catch a Brute or Leviathan Pod using Kinesis
I've never had a brute shoot at me, personally, so I did this during the Leviathan battle in Chapter 6. Wait until the mouth starts spitting pods at you, grab it with kinesis, and shoot it back.
Here's how to do it on a brute, stasis it run around and shoot off one of his legs, instead of charging he'll just shoot pods at you, catch them and shoot them back. Credit to Con818 for this tip.
Exterminator
Kill the Leviathan.
Yet again, you can't miss it. This is the boss at the end of Chapter 6 and is very easy. Simply shoot the tentacles in the yellow spots, and shoot the mouth when it opens. Simple as that.
Get off my ship!
Kill the Slug Boss.
Once again, you can't miss it. This is the second cannon segment on Chapter 8 where you are fighting, well, a giant slug. Just shoot the tentacles in the yellow spots and shoot down the trash they throw at you.
Slugger
Kill the Slug Boss with more than 50% shield strength remaining.
This is much easier than its predecessor. Just take it slow, concentrate on one tentacle at a time, and don't take any unneccessary chances.
Mindless Prey
Kill the Hive Mind.
Ah, the final boss of Chapter 12. Despite its size, it is easily taken down. To begin with, strafe from side to side to avoid its tentacles then shoot the yellow spots on its face. Lather, rinse, repeat. When two are left, it will pick you up, and you must shoot at it while hanging upside down. There is a time limit here, so be careful. After it drops you, return to the strafing loop, but this time, shoot at its exposed chest. Lather, rinse, repeat, Trophy Unlocked.
Kickin It
Escape from a Lurker's grab attack 10 times.
To pull this off, you have to be grabbed by a Lurker 10 times, then fight them off by rapidly pressing the X button. Lurkers are the baby Necromorphs with three tentacles. Easiest/Quickest way to get this is whenever you see one don't shoot it, run up on it until it grabs you. Here is a picture of what the enemy looks like to avoid any confusion: