In Chastain Heights, south of the movie theater, there is an apartment building. Observe the retractable ladder on the fire escape. If you have a hook, use it to pull the ladder down. Otherwise, run west across the street, and find another hook leaning against some red garage doors. Pick up the hook and use it to bring down the ladder.

Climb up the ladder and enter the building through the window. Near where you enter, you will see a Gramophone (Phonograph). The open inventory prompt comes up, but at this time you don't have the item you need. Time to search the apartment.

First, climb up the stairs, turn left, and go into the baby/child's room. Pick up the crank from the crib. You receive:

Crank
An antique-looking hand crank

In the bathroom next to the kid's room, there is a first aid kit.

Next, go to the master bedroom across. There is a bed inside covered with blood. On the dresser near the door, find:

Vinyl Record
An old gramophone record

Go back to the gramophone downstairs. Use the crank on it first, and then put in the vinyl record. Start by turning the crank to the right (clockwise), then stop and survey the room. Some words appear on the floor, next to the lady's corpse. It says: "TURN BACK TIME". This is a clue to help you solve the puzzle.

Silent Hill: Downpour Gramophone Side Quest

Gramophone Puzzle Solution
You need to play the gramophone BACKWARDS. Rotate the crank left (counter-clockwise). Watch a crime unfold backwards, depicting a male figure murdering a woman with an axe. Your objective now is to put the victims' spirits to rest. Note the two portraits on the wall: one male, and one female. There will be a lot of blood pouring from the female portrait. Out of the the male portrait, the spirit of the murderer emerges. You can attack and kill it, but it just causes him to re-emerge from the portrait. The answer to this puzzle is to use your LIGHTER to burn the male portrait down (On easy and normal puzzle difficulty levels, there will be an audio clue consisting of fire burning, hinting at what needs to be done).

You get the trophy/achievement:

Turn Back Time
Completed "The Gramophone" side quest