Materials you will need:

1x Plain White T-Shirt
Matte Black Fabric Paint (Amazon)
26x Addressable RGB LEDs (Polulu)
Solder, and Electrical Wire
Heat Shrink Tubing (Maplin)
1x Arduino Uno
1x USB Battery Pack
1x USB-A Cable
1x Needle & White Thread

Equipment you will need:

a Soldering Iron

Step 1: Soldering the Data Lines

The first step is to construct the chain of LED lights. If you're using different LEDs to Pololu ones used in this project, you will have to make slight modifications, but the principle is the same.

Soldering

We'll start by soldering the data lines together. For all 26 LEDs, the DOUT pin needs to connect to a DIN pin. The last LED in the chain will be left unconnected, and the first LED will need a long wire that will eventually connect to the Arduino.

Heat Shrink Tubing

Since the LED pins are close together, we will need to apply heat shrink tubing to make sure they don't touch as they're moving around on the T-Shirt. We will need to add these to the wires now, but we will not shrink them till all the wires are soldered.

Things to Remember

The Stranger Things wall has a 8-9-9 configuration, so remember to make the wires longer where necessary
Be sure to add two bits of heat shrink tubing to each digital wire, and try not to get the soldering iron too close to it, so it doesn't inadvertently shrink
Be especially careful to correctly identify the DIN and DOUT pins. The DOUT pin is significantly larger, and lines up with the flat edge on the side of the LED

Step 2: Soldering the Power Lines

Next we need to solder all the ground and power lines. This follows the same process as the data lines, including the need for heat shrink tubing on each connection.

Each GND and 5V leg on the LED will need two wires connected to it, one to the LED before and one to the LED after (so the last LED in the chain will just have one wire connected to each leg).

Step 3: The Heat Shrink Tubing

Now that all the cables are soldered, we can shrink the heat shrink tubing. Before proceeding it's worth plugging the 5V and GND lights into an Arduino to ensure everything is connected correctly. If all the LEDs light up, you're fine, if not double check the arrangement of the LEDs to ensure you haven't soldered one the wrong way round, or got the 5V and GND pin confused.

If everything checks out, you should be able to shrink the tubing with a heat gun, or a simple lighter.

Things to Remember

Run the heat source quickly over the heat shrink tubing, back and forth to make sure no one spot gets too hot
Be as careful as you can not to heat the LED, as this could damage the component

Step 4: Programming the Arduino

Make sure you have the latest Arduino programming app installed on your machine.
Navigate to Sketch -> Include Library -> Manage Libraries...
Search for PololuLedStrip and install it

Once that is installed you'll have a few examples test your LED Strip. Navigate to File -> Examples -> PololuLedStrip and pick an example at random. At the top of the code that appears, you'll see this line:

PololuLedStrip<12> ledStrip;

Connect the 5V wire to the 5V port on the Arduino, the GND wire to the GND Port, and the data wire to pin 12 (or change the 12 in the line of code above. Click run, and your LED strip should light up.

Step 5: The Code

The code I used is available on GitHub.

A lot of this code goes beyond what you'll want to do, as my T-Shirt also includes a hidden switch which can change States, and display different animations. Feel free to fork the project and add your own animations.

For the purposes of the Stranger Things wall, this is all you'll need.

The only caveat is that the default code says the phrase "MERRY CHRISTMAS". To change this to whatever you wish, navigate to this line of code:

char text[] = "M E R R Y C H R I S T M A S ";

Change that to any text string you like, but make sure everything is in capital letters, and each character has a space between it, as this will provide the unlit gaps between letters in the animation.

You then need to navigate to the sequenceCount function, and change the number 32, to the number of characters in your defined string (including spaces).

Re run your code, and your T-Shirt should light up with your new message.

Step 6: Sewing

You can now sew the the LED onto a t-shirt. The Stranger things wall has a bit of a messy look, so neatness is not a priority at this point. I added to this look by twisting the wires together, as pictured before.

If you've never sewn before (like me in this project), have a look at this video. You will essentially be sewing round and round the wire till you're satisfied, to keep it in place. Thankfully, the messy nature of the wall means that poor sewing technique can also be forgive.

Step 7: Painting the Letters

The fabric paint I chose to use in this project comes with a nice thin spout, making drawing easy. Start by placing some card on the inside of the T-Shirt, as the paint will go through the fabric. Using a screenshot from the show as a guide, carefully paint the alphabet next to the letter's corresponding LED. The typeface isn't too difficult to mimic, and as mentioned before, a slightly messy look actually comes out better than a clean one for this project.

Step 8: Finishing Touches

We now need to attach the Arduino to the T-Shirt. Doing this makes it much easier to wear, as there only needs to be a single USB cable coming from the T-Shirt to an external battery housed in your pocket.

Start by cutting a small hole in the side of the T-Shirt and feeding each wire though. You can then take the Arduino and sew it on the INSIDE of the T-shirt. Place it towards the side, so that it is not visible, and won't compromise you sitting down. Once sewed, cover it with some kind of tape, so that the pins stay in place, and the Arduino doesn't press into your skin uncomfortably as you wear it.