Head east on Rice Street. Close to the Spillway, a couple of houses before King Ave, on the south side of the street, there is a house. If you have the UV Flashlight you can see footsteps on the pavement leading to the back part of the building. Follow the narrow alley around the house, and enter the door. See the Silent Hill map below if you are having any problems locating the start point of this side quest (Start point labeled with red "X"):

Dead Man's Hand Side Quest Silent Hill Downpour

Explore the inside of the house. One of the rooms has a poker table with cards on it. One hand is turned face up: Eight of Clubs, Eight of Spades, Ace of Clubs, Ace of Spades. Aces and Eights. Two black aces and two black eights. The Dead Man's Hand.

Open another door, leading you into some sort of utility room. There is a body on the floor. Seems like his heart was ripped straight out of his chest. This starts side quest, Dead Man's Hand, and your objective is to find the missing heart.

"Found another prisoner. Something tore his heart out... so why can I still hear it beating?"

There are stairs leading to the basement near where the corpse is. Go down to the basement. Explore the area, there is a fireaxe if you need a new melee weapon. In another small room, find a hole with a ladder going down. Climb down the ladder to end up in some underground corridors. You now need to find the heart.

Follow these instructions exactly that will guide you through the maze and towards the heart (On Normal and Hard Difficulty):

From the ladder, go straight towards the far end
Turn left, going through a gate
Turn right
There is a small intersection straight ahead. Go past it
In the second intersection, turn right
Keep following the main corridor to the end, open a gate and you will find the human heart.

If you get lost, go back to ladder and follow the instructions again to get through this labyrinth of underground corridors.

On Easy Difficulty, the heart will be located in a different place. Use your UV Flashlight and follow the tracks to and from the Human Heart.

You get:

Human Heart
Heart ripped out of a man's chest

There are a couple of Screamers on the way to the heart. But once you pick up the heart, a lot of Screamers, and some Weeping Bats, appear in the corridor. Make a run for it, retracing your steps back towards to the ladder. It is not recommended you try and fight, as there are too many of them. Also be careful not to get surrounded.

Climb up the ladder, exit the basement, and return to the heartless body. Use the human heart on it. You will put the heart into his body.

You receive the following trophy/achievement:

Telltale Heart
Completed the "Dead Man's Hand" side quest.

The body disappears, and transforms into a first aid kit. This completes the Dead Man's Hand Side Quest.