The basic Halloween Jack is just the a tattoo of his bony frame draped in black clothing and his head, either wearing a black top hat or nothing at all. For this basic design, choose your favorite moment in the film, when Jack is still in Halloween garb, and have your artist develop a stencil based on the image. It can be while he's singing a song or while making a devilishly frightening face. Consider green highlights to show the rotting texture of his dead skin and/or orange to continue a general Halloween theme.
Choose a quote or line of dialogue from the film, "The Nightmare Before Christmas," either spoken by Jack or about him, and work with your tattoo artist to develop a bony, Gothic claymation-style script for it. The more detailed the font, the larger the letters will need to be, so try to keep the quote limited to only a few words -- this may require only using a small portion of the quote. Also, consider incorporating Jack's head into some area of the script to reference the quote.
For a portion of the film, Jack dresses as Santa Claus, with a full red ensemble with hat and white beard to match. This image of a Christmas Jack is much cuter and sweeter portrayal of the character Jack Skellington tattoo that isn't dark or ominous. Consider either tattooing his entire body in Santa regalia or just his head with the red hat and white beard; both will portray the same positive image.
Yin yangs are classic designs of opposing elements, molding together to form a single image, usually a circle. A yin yang Jack is a fresh take on the traditional image, using Jack's face as the basic shape and each side is adorned with the two opposing characters Jack is in the film. One side should be the somewhat frightening and ghoulish "Halloween Jack" with black top hat, while the other side should have a white beard and red hat of the "Christmas Jack."