"Pumpkin King" is Jack Skellington's title. It recognises him as the King of Halloween. He is the one who travels to the real world every Halloween to scare people.
There are two possible reasons Jack is referred to as the Pumpkin King despite being a skeleton.
One is that the title states that the bearer is the scariest creature to exist, something which has been hinted at Jack being throughout the entire film.
Another possibility is that Pumpkin King refers to Jack in his wicker-man or scarecrow costume with a pumpkin for a head. The opening scene of the movie shows Jack as a wicker-man attached to the town sign, which then twirls around and seemingly points the way to Halloween Town. Towards the end of "This Is Halloween", the wicker-man is shown being wheeled into the center of town on a wooden horse. He appears to be just a stiff bundle but then jumps to life and grabs a burning torch from Helgamine, the tall witch. He then swallows the torch, setting himself ablaze and then flails around in a fiery display before jumping into the air and into the town's fountain. A few seconds later he ascends from the fountain as his normal self. It is assumed this is a custom that happens every year to celebrate Halloween.