There are 18 achievements to try for in this game. Some of them you'll get just by going through the game, but some take special effort, and there are several that can't be combined. If you want to get them all, you'll need to play through the game at least twice. If you're still on your first play-through of the game, beware of spoilers below!
Teamwork: Solve a puzzle together. You'll get this one during regular game play by solving a puzzle using both Joey and Rosa; the first time this happens is during the Mary case.
Curtain Call: Save your first ghost. That would be Mary.
Dumpster Jumper: Make a leap without falling once. When you're in the alley behind Vantini's, jump from the dumpster to the fire escape ladder on the first attempt. To do this, make sure you melt the ice on the dumpster and wipe off the resultant sludge before you climb onto it.
Ghostly Silent: Keep clam [sic] and keep cool. During your interrogation, respond to every question with silence.
Online Troll: Gain access to an online game. You get this one when you hack into Kendra's Trollgate account.
Legacy: Reunite a family. You get this one when you bring Peter and Maggie together.
Unbound: Free yourself. You get this one when Rosa gets out of the straitjacket at Bellevue.
Convergence: Two becomes one. You get this one when you bring Heather and Tanya together.
Deception: Try to lie your way out of it. Rosa is frequently required to explain herself, and you'll get this one by always going for a lie rather than the truth. Here are people to lie to: Kendra, the clerk at the Unwired Abacus, Lia, Officer Cory during your interrogation, Ray when you're caught in her office at Vantini's, and Joe Peebles at Channel 11. Some of these are pretty mild lies, such as telling people you work for the police when you actually work sort of parallel to them.
Epiphany: The hows and whys of Joey. Learn everything you can about Joey's past. In addition to the two flashbacks that happen in the course of the game, make sure Joey talks to Madeline after she takes over Rosa's body and before you evict her.
Time to Move On: Complete the story. To win this one, finish the game at least once.
Move on and Listen: Finish the game in commentary mode (from the first screen to the last). Turn it on, start a new game, and play it through to the end without turning it off again. The commentary is pretty interesting if you want to learn more about the creative process; if not, well, you can always just click through the actual comments. Another reason to turn it on is that at one point in the game (just after Rosa's arrest), if you have the commentary on, you'll be given the opportunity to play a deleted scene. I love that kind of junk, even when I don't agree with his creative choices.
Blowey Mallone: Blow on everyone you can find. In every scene where you have control of Joey, try to use his wind power on everyone, ghost or human. There are some he won't blow on, but try everyone all the same.
Goldfarb Digger: Find a secret room. For this one, have Joey use his tie on the painting at Vantini's. What you have here are six characters Abe Goldfarb has voiced in Wadjet Eye games interacting with each other. You can't interact with them, so you can either sit and listen until they start to repeat or leave immediately via the vortex (the white light in the sky at the left).
Softly Softly: Create no more than 450 footprints in the snow. (In some versions of the game, it's 300.) To keep Rosa's footprints to a minimum, don't walk around when you're outside, use Joey whenever you can, leave every scene by the nearest exit, and when you can, finish the outdoor scenes in one or two visits. Watch the end credits to see how many snowprints Rosa left this time.
Snow Plow: Somebody's got to do it... Pace back and forth about 7 to 10 times in every snowy area until Joey gets annoyed and tells you to stop. (This only works in outdoor areas; in the snowy second floor of the Karth Building, Rosa can tromp about for half an hour and Joey never complains.) If you want to try for this achievement, make it a point to do this the first time you enter an area; after you clear an area, it will disappear off your map and you can't return. Don't forget to do the roof next to George's apartment and the police station after your arrest (before that, Durkin comments rather than Joey, which doesn't count).
Laying It Bear: Learn a long-kept secret. When Rosa is in the void with Lauren in the endgame, talk to Lauren after you convince her to get up but before you climb the ledge. This gets her to tell you about the teddy bear she gave you.
Joined the Music: She has plenty of time ... To get this one, save Kendra.