Step 1: Beginning Steps

Role your cardstock into a cone shape and tape it so it won't unravel. This will be your gramophone that amplifies the sound coming from the record.
Take the needle and poke a hole on through the small end of the cardstock cone and tape or glue it there.
Take your box and flip it where the bottom is facing up. Then poke a hole on the short side of the box and make sure your stabilizer will fit in the hole and protrude at an angel. Glue it in place and to the large end of the gramophone.

Step 2: Next Steps

With the needle protruding from the small end of the gramophone, make sure the sharp end of the needle is facing down toward the record.
Now you will need to take your pencil and put it through the hole in the center of the record while making sure It is secure to where the record won't slide around on the pencil.
Finally you will need to poke a hole in the top of the box, opposite the stabilizer. The record will need to be about 2 centimeters away from the box.

Now all that is left is to place the needle on the record.
Take hold of the end of the pencil and spin it in the desired direction. This will make the record turn, creating your music.
Voilà! you have a working record player that cost's little to nothing in money.
Depending on your style or availability of the supplies this record player can be adjusted. You could also add a motor to make the record spin. This particular record is a 33 so I would need a motor with 33 RPM.