To make your door more soundproof you need to seal it properly a good method od achieving this is to use magnetic door gaskets.
Build soundproof door.
Install a soundproof weatherstripping door kit weatherstripping products are meant to address the gaps between the door and the door frame or casing.
Wait until the adhesive is dry and your door will be soundproofed.
Doors have to be visually appealing and hence the traditional wooden materials cannot be replaced with soundproof materials like drywall and brick.
These do the same thing as weatherstripping they seal the door frame but they do it far more efficiently and effectively.
Attach the vinyl to the door using a construction adhesive which you can purchase at your local hardware store.
Door gaskets are commonly used on fridge freezer doors to ensure an airtight seal.
The self adhesive rubber foam tape should compress when you close the door creating an airtight seal all around.