Mainly, this novel is narrated in the third person; that is, the story is usually told as "he said" or "she said" or "Scrooge watched them," etc. In the beginning, though, there is a little touch of a first-person narrator, as someone talking directly to the reader, referring to himself as "I." This narrator is the type of personality who will use a phrase and then mull over its appropriateness ("I might have been inclined, myself, to regard a coffin-nail the deadest piece of ironmongery ... ") and to make humorous satirical remarks.
This first-person voice fades away once the characters in the book start interacting with one another, leaving the characters and the action of the novel to keep the readers' attention. The last time this first-person narrator is heard from is when it remarks on how strange it is that Scrooge, who had not thought of Marley since hours earlier, would see his face on the door knocker ("let any man explain to me, if he can, how it happened that... ")