There are two streams of narrative in the Santa Claus story, and thus two "appearances".
The first, of course, is the historical. Not exactly on-topic here, but you can research the historical figure, Nicholas of Myra, who was a bishop of that city in the third century. Known for his charity, he was recognised as a saint by the Church. Historically, he was obviously "born as an ordinary man" and would similarly have "appeared" as an ordinary man. Some folks measured & studied his remains and came up with what they think he might have looked like.
The other stream, of course, is the legendary & mythological. The midwinter bringer of jollity and giver of gifts. This character is based on the earlier St. Nicholas matter. His iconography (red cloak, peaked hat and a staff) are all the standard signs of a Catholic bishop, which St. Nicholas was; and his modus operandi (secretly bringing gifts) seems also to be based on St. Nicholas's history.
Various early characters of the 15th through 18th centuries would help to give Santa Claus his basic appearance in Western culture -- Father Christmas, Sir Christmas, Lord of Christmas, etc. But the popular, modern appearance seems to have been founded upon the poem “A Visit from St. Nicholas by Clement Clarke Moore. It is here that Santa Claus per say appears out of the earlier Christian matter. Moore combines the Catholic saint with early American Dutch traditions and some earlier Odin mythology to create St. Nick anew.
As the decades went on, other stories and attributes got intertwined with the older figures. Santa Claus was found to have his workshops at the North Pole. He got his great book, listing all the children who have been naughty or nice. For quite a while, Santa appeared in a variety of costumes. Some were the more traditional garb of a bishop, others were more along the lines of a tinker or craftsman. Sometimes his great cloak was green or red or yellow or even patriotically star spangled. His size & physique too were quite changeable: sometimes he was quite small, other times he was quite tall; he's been thin and fat and comfortably round.
The modern image wouldn't be cemented until the early 20th century: red, fur lined coat, red trousers, black boots, red fur lined peaked cap, with or without a pipe.
As far as works go, you can't go wrong in looking back to Moore's poem. Be on the lookout for the images created by Thomas Nast in the 19th century. And of course, the Life and Adventures of Santa Claus by L. Frank Baum. Here you'll find excellent explanations and origin stories for many of his attributes, starting with his birth & childhood and going right up to the time he was granted immortality.