I'm not sure how much of an instructables this will be: I will give you the circuit and show you my finished project, however this project is like painting a picture: it's subjective to your canvas and your style.
Original: CLICK PLEZ PLEZ CLICK ME.

So, I made two of these. I like the second one better than the first.
Here are videos:

First:
Second:

Look at the pictures too!

 

Circuitry

So, I would run the TIP31 IC on 9 volts or higher. 
You connect BASE to your audio source
You connect EMITTER to ground
You connect COLLECTOR to the end of the LEDs.

Basically, when the base has a voltage, the transistor allows the cathode of the LEDs to be connected to ground. The collector connects to the emitter.

So, I would estimate that each LED is about 3v, 2mA. Depending on your LEDs you want to use, you will need to measure each one.

When you connect LEDs in parallel, the voltage needed to drive both will be the same, but the current will be the sum of the two.
When you connect LEDs in series, the voltage needed to drive them will be the sum of them, but the current needed to drive them will remain the same.

As you can see in the picture, I have 6 LEDs, two in parallel, so I need 5*3 volts to drive all of these.

I decide that 12v is close to 15v, so I chose to use a 12v power source.


Now before you move onto the next step, please plan out your circuit and maybe test it on a breadboard before you do something irreversible.

 

Construction

I was bored one summer, which is why I decided to make this. I had a TON of time. 
The hardest part about this is having patience and planning.

First, you might want to decide where you want your circuit: inside the car, or outside?
If you have an extremely detailed car (like the ones you need to build the engine and glue stuff), I would recommend putting the circuit on the outside.

If you have an easy build model car (like the snap together ones that you could build in 5 minutes), I would sacrifice some of the detail and plastic to fit in a circuit.

Now that you figured out where you want your circuit to be placed, I would recommend using small batteries for internal circuitry. I used an A23 battery cell for my second car. That worked GREAT. For my first car, I used an AC adapter, and being old, the AC adapter power surged and burnt up most of the components :(


So, build the circuit (and make it compact)
place it in where you want it to be (In the pedestal, or in the engine, for example)
Drill holes where you want your LEDs to be (base, engine, carriage)
Glue in your LEDs (thats right, glue them in place before you do anything)
Solder wires connecting your LEDs according to your circuit.
Connect it all back to the TIP 31!
 

Advice

I like to have an audio output too, so I had a speaker on my first car
On the second car, I could have added a speaker inside the car, but the audio quality was terrible. I just replaced the exhaust pipes with two audio jacks: one input, one output.

When drilling the holes into your car, be careful. You want the car to be disassembled when you drill one part, so you do not damage anything else.

Check polarity!

Don't melt the plastic with your soldering iron

Be creative