For this achievement you simply have to finish the first episode of the game, As soon as you wake up you'll notice you're in the jungle, there's a woman not far from you, find her and follow her through the jungle, she'll lead you to Kate, when you speak to Kate you'll trigger a flashback, this is where you get the Solution for 815 in Lost: Via Domus
, when you're done in the flashback talk to the hostess and you'll find yourself back in the jungle with Kate again, talk to her once more and she'll point you in the right direction, head in the direction she pointed you and you'll come across a dog that you need to follow, you'll need to hold down "RT" to run and keep up with the dog, after a short time you'll find yourself on the beach, when you arrive at the beach you need to locate Jack, he's dressed in a black suit and will be attempting to revive another survivor down on the shore, when you speak to him he'll ask you to stop the fuel tank leaking, this is where you get the Solution for Save the castaways in Lost: Via Domus
, once you've done a short cutscene will play where Jack thanks you and then it'll be night time, to finish the episode you have to examine a backpack on the beach.
I've also linked in a video which will show you all of what I've said and more.* Credit to 360VideoGuides for the video