There are currently 9 achievements for this DLC. I highlighted the three acts to make it clear at which point you can get each achievement. You may need to do additional steps in previous acts for getting it so pay attention from the start.
You can get all achievements in 3 walkthroughs from beginning to the end. Or you could replay couple of episodes.
You can't "save" them all! In 2nd act you eventually need to give to the ghost 3 out of 4 victims. The first "killed" victim will end up in the Vault. The last survivor - in the Projection room. Keep this in mind for successful completion.

Don't Touch Me!



Fastest to get. When Diane calls and tells you not to mess with SpiritJammer, you need to do exactly opposite. Just flip any switch and listen to her reaction.


A Wallet Full Of Treasure
Determine the fate of Brent Backflip.


Choose the Stage (with megaphone) as third room (the previous choices don't matter).


A Possession To Remember
Establish Cecil Sloane's perilous past.


Choose the Prop Store (with dumbbell) as third room (the previous choices don't matter).


A Fall From Grace?
Discover the cause of the falling stunt man.


Choose the Costume Room (with hanger) as third room (the previous choices don't matter). You should get the first restored tape here.

A Burning Question
Solve the mystery of the unfortunate fire.


Here you need to finish this act with Patrick in Projector room and Holly in the Vault. Patrick must be saved and eventually be the last survivor if you saved someone else. Holly must be your first "killed" victim, so save any victim of ghost attack before her. The other choices don't matter as long as these two conditions are met. You should get the second restored tape here.

I Think I Felt A Chill
Complete the Live & Spooky Bannon Special.


Just get any ending at the final of Act 3.


A Deadly Comeuppance



For this ending you can pretty much relax and don't do anything with SpiritJammer. As long as your first "killed" victim is not Holly, you're okay. Wait till the final to get achievement.


Close But No Cigar



You should not solve first mystery. In the 1st act choose any room except Costume Room as your third choice.
You need to finish 2nd act with Holly in the Vault (thanks to redotheundo for pointing this out in comments). Holly must be your first "killed" victim, so save any victim of ghost attack before her. The other choices don't matter as long as this condition is met.
Wait till the final to get achievement.


Credit Where It's Due



Here you need to restore all 3 tapes.
In the 1st act choose the Costume Room (with hanger) as third choice (the other choices don't matter) - you'll get the first tape.
You need to finish the 2nd act with Patrick in Projector room and Holly in the Vault. Patrick must be saved and eventually be the last survivor if you saved someone else. Holly must be your first "killed" victim, so save any victim of ghost attack before her. The other choices don't matter as long as these two conditions are met. You'll get the second tape.
In the 3rd act Holly collects last tape automatically. Wait till the final to get good ending and achievement.