Find Nathan Gould
Story related. Received during chapter 5, Lab Rat.
Kill 10 enemies by throwing an object at them
Just about everything in the game can be picked up and thrown, but killing enemies with them is another story. To make things easier on yourself, every time you see the Ticks scavenging about, pick something up and throw it at them. Nail ten and you will have this easily.
Assist the evacuation at Bryant Park
Story related. Received during chapter 12, Train to Catch.
Food for Thought
Kill a CELL operator with a giant donut in Lower Manhattan
In the mission Second Chance, you will come to a rooftop that has a Large donut on it to the left. Tag the enemies below. See where they are through the wall and then wait for them to go in front of the donut. When they are in position, give the donut a push and cause the death of the soldier below. Congratulations, you're now responsible for the most hilariously sad letter to ever be written home to the wife of a soldier.
Escape the Battery Park evacuation center
Story related. Received during chapter 2, Second Chance.
Single Player: Kill 4 enemies in a row with headshots
When you headshot an enemy, you will get a counter that says 1/4. Keep the tally going will 4 consecutive headshots and you will nab this. Easily done in stealth mode by popping one enemy and running away to get a stealthy shot on another, etc.
Heart of Darkness
Complete 6 levels on Supersoldier difficulty
This is the toughest of the tough. The easiest way is to play through on either Recruit or Soldier first so that you can upgrade your suit to its max. The upgrades carry over through playthroughs so the Supersoldier playthrough will allow you to use those upgrades.
Hole in One
Throw an alien down the sinkhole in Dark Heart
After the second pillar you destroy during Dark Heart, you will come to the area that houses the third pillar. When you enter it, to the left over the edge will be a whirl pool. Grab an enemy and give them a toss into the hole and watch them vanish for the achievement/trophy.
Reach Central Park
Story related. Received during chapter 19, A Walk in the Park.