How is Ebenezer Scrooge of A Christmas Carol a Gothic character?
The character of Ebenezer Scrooge is one of the Gothic elements of Charles Dickens' novella, A Christmas Carol. Characters in Gothic novels are often one-dimensional, or stock, characters. We know what to expect from such characters because we have seen them often before, and they do not change over the course of the novel. This, of course, is where Scrooge as a Gothic character breaks down--because he changes dramatically as a result of his experiences in the novella. However, putting that aside, we can see that at the beginning of the novel, he has definite Gothic characteristics. Often Gothic novels feature a character who is a tyrant: He rules with an iron fist, he is unapproachable, and he is extreme. Scrooge is such a tyrant in Stave I. He keeps the office so cold that his clerk, Bob Cratchit, has to warm his gloved hands by the candle as he works. He scares away the Christmas carolers who come by, and he gruffly brushes off his nephew's kind invitation to Christmas dinner. The statements he makes to the men who come to ask for charitable donations show an evil heart; he suggests that those who are poor and ill-fed should simply hurry up and die and "decrease the surplus population." Scrooge then goes to his home, which although it is not a Gothic castle, is definitely reminiscent of one, being in a lonely, industrial part of town, very dark and foggy, and very old and sparsely furnished. When Scrooge meets Marley's ghost, who wears "the chains [he] forged in life," we learn that Scrooge's chains, which he has been forging seven years longer, are already heavier than Marley's, if we could but see them. Even Scrooge's physical description paints the picture of a terrifying tyrant: "The cold within him froze his old features, nipped his pointed nose, shrivelled his cheek, stiffened his gait; made his eyes red, his thin lips blue; and spoke out shrewdly in his grating voice."
While A Christmas Carol might seem Gothic at first, with its abundance of ghosts and clanking chains, Scrooge doesn't seem to be a Gothic protagonist, not in the traditional sense. Protagonists of Gothic novels tend to be avatars of innocence, and as such are very often young and female. Examples are Jane Eyre, Emily St. Aubert of Ann Radcliffe's The Mysteries of Udolpho, and Laura of Carmilla. These hapless young women are usually plagued and victimized by powerful evils out of their control, which in their turn are almost always metaphors for oppression, hence subgenre designations such as Female Gothic (Jane Eyre, Wuthering Heights) and Irish Gothic (Carmilla). If these protagonists get happy endings, which they don't always, it's by the power of their innocence and goodness.
I say all this because when A Christmas Carol is viewed in this light, Scrooge begins to look a lot like a Gothic villain, representing the evil of avarice. He's the consummate miser at the beginning of the story, and just letting Bob Cratchit stay home on Christmas Day is like pulling teeth for him. Furthermore, the Ghost of Christmas Present implies that Scrooge's unmitigated greed will be the ultimate cause of Tiny Tim's death.
Which brings me to a related point: if Scrooge is the Gothic villain in this story, the evil powerful oppressor, then Tiny Tim and Bob Cratchit represent two different facets of the Gothic protagonist. Tiny Tim is obviously the avatar of innocence, like I mentioned earlier, and Bob Cratchit represents the virtuous, working hero who becomes more common in later Gothic fiction (e.g. Robert Audley of Lady Audley's Secret, Gabriel Utterson of Jekyll and Hyde).
Back to Scrooge, though. To provide a short answer to your question, Scrooge absolutely starts out as a Gothic villain, and he's redeemed in no small part due to the innocence and purity of Tiny Tim and Bob Cratchit.
I hope this helps!