I mentioned that all the achievements are story related, so there is no excessive searching for collectibles, beating a chapter in a certain amount of time or kill requirements. Simply play through the game and you'll unlock these.
The first achievement you'll earn (after a small bit of dialogue with a cop) is when you get into an accident in which you black out, whilst the cop is killed by Walkers. Eventually, you'll gain control of Lee, have him kick out the car window and limp over to the dead body. Use the crosshair to find the keys (on his belt); once the cuffs are off, you'll be attacked by the dead cop. Use the control stick to back away until another small scene where Lee sees a shotgun shell. Pick it up and turn to the left to grab the gun, place the shell into the gun (eventually) and kill the Walker.
After escaping the horde of Walkers, you'll come to a house (this is where the demo starts), enter the house and either look around (examining the TV, bookcase, kids painting), or simply investigate the answering machine (there are three messages from Hanna in Savannah), continue into the kitchen where Lee will slip on a puddle of blood and hear a voice out, 'Daddy?' Examine the drawers until you find a walkie-talkie, this will introduce you to Clementine, and share some dialogue with her.
You'll get attacked by a Walker, and slip in the blood once again, slamming your head into the counter, causing blurred vision. Move the crosshairs to the zombie and hit 
The Walker will trip you (and by now you should have come to the conclusion that Lee lacks the ability to stay on his feet), and Clementine will appear with a hammer, grab it from her and kill the zombie, netting the first achievement. Continue until you meet Shawn and Chet, help them get their vehicle moving and you'll earn the second achievement.
When you arrive at the Farm, Hershal will ask you a bunch of questions about who you are, how you know Clementine and where you were going, answer them and retire to the barn. In the morning, Kenny will wake you up and talk with you a bit. Continue talking to him and ask him about his plans, once you're done talking with Kenny, talk with Clementine and Katjaa. Walk around to find Shawn working on the fence with Duck nearby; help saw some wood until Shawn tells you to go see his dad in the barn.
Talk with Hershal until you hear Shawn shouting, this is the first of two major decisions you'll have to make in this episode: save Shawn, who's pinned under the tractor; or Duck, who's being grabbed by Walkers, this will net the achievement for you. It really doesn't matter who you choose as Duck will ultimately be saved.
Once you arrive in town, you'll get ambushed by Walkers, but be saved by a girl named Carley, who'll tell you to get to the Drug Store they've taken shelter in. An older man, Larry, will accuse Duck of being bitten and try to have him thrown out, you can defend him by siding with Kenny, but it doesn't matter what you choose to do, as he will be saved. Clementine will get attacked while trying to use the restroom, kill the Walker.
Talk with Carley, who is having trouble turning on a portable radio. Examine it and turn it over, open the battery slot on the back and you'll learn why it can't be turned on. You'll find one battery near the greeting cards, between the door to the Drug Store and Kenny, Duck and Katjaa, and the other near Lilly and Larry.
Head into the Employees Only area and head to your right after coming in, the camera will reposition itself and you'll see a plank covering a desk. Move it out of the way, while talking with Clementine, until she gets a small cut. Head over to the first-aid box that is conveniently placed in the same room and grab a bandage. Once she's taken care of, examine the desk drawer to find a Universal Remote, which you'll need later. There is also a picture on the floor near the cot that will explain a lot about who Lee is, if you can see the picture well enough that is.
Upon exiting the back room, Glenn will call via walkie-talkie saying he's in trouble, which will cause Lee and Carley to go to his rescue. Once you arrive at the motel, and find Glenn, he'll say he heard a women crying and wants to rescue her, to which Lee, and to a lesser extent Carley, agree. When you retain control of Lee, lean to the left and grab the pillow. Move to the right, which will place you on the side of a truck that has an ice pick inside, but no means of entering without creating noise. Look ahead and you'll see a zombie against a car; use the pillow on him as a silencer so Carley can shoot him. Once you regain control, open the car door to find a spark plug, and put the car in drive, trapping another zombie. Return to the truck and use the spark plug on the window, which will allow you to grab the ice pick. Return to where you started, this time advancing to get behind the van, lure the zombie over from the side and stab him with the ice pick, then kill the one you trapped and grab the axe. Once you go upstairs, kill the two zombies and break open the door, you'll find the woman, who has been bit. Give her the gun or no, either way, you unlock the achievement.
Very easy if you grabbed the Universal Remote earlier, but in case you didn't, head into the Employees Only area, and head to your right after coming in, the camera will reposition itself and you'll see a plank covering a desk. Move it out of the way, while talking with Clementine, until she gets a small cut. Head over to the first-aid box that is conveniently placed in the same room and grab a bandage. Once she's taken care of, examine the desk drawer to find a Universal Remote. There is also a picture on the floor near the cot that will explain a lot about who Lee is, if you can see the picture well enough that is.
Talk to Doug and say yes to going outside, talk with him if you want to learn a bit more about him, but otherwise, examine the Walker trapped under a telephone pole, to which Lee will say that he has the key to the pharmacy. Use the axe on the lock and pick up the brick, throw it at the shop across the street and try to use the Remote. Doug will get it to work, being the nerd that he is, and give you time to grab the key. You'll learn that the Walker is actually Lee's brother, and being the brother that Lee is, he decides to put him out of his misery.
After returning to the Drug Store and getting the pills for Larry (not that he deserves them), you'll trigger an alarm which, as you may have guessed, lures a bunch of Walkers to your location. You'll have to hold the door closed, since you decided to break it (nice move). Enter the commands as they appear until you are forced to make that second big decision, to save Doug or Carley, it doesn't matter who is saved. Clementine will be attacked, enter the commands to save her. Run towards the door only to get punched in the face by Larry (see, he's an asshole), but Kenny will come to the rescue, grab his hand and you'll earn this achievement.
Simply talk with everyone in the parking lot and you'll earn this final achievement.
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