How does Arthur Conan Doyle present the character of Thaddeus Sholto?

Thaddeus Sholto is an eccentric, complex, fearful, and rather unpleasant person whose actions set the story in motion. Initially, he is guilty of conspiring to keep a fortune away from its rightful heir, Miss Morstan—a young woman who must struggle to make a living. Yet his conscience bothers him enough to secretly and anonymously send her a kind of guilt-payment each year—a package containing a valuable pearl. He finally sends a mysterious summons to Miss Morstan, which prompts her to seek Sherlock Holmes's advice.
In addition, when the murder is discovered in Chapter V, Thaddeus is intimately connected with it. The dead man is Thaddeus's twin brother and fellow heir to the creepy estate of Pondicherry Lodge. The police arrest him for the crime.
So Thaddeus is important to the plot, but there is more: His introduction to the reader also foreshadows many of the key themes of the novel—selfishness, greed, dishonor, betrayal, revenge, and making amends. Conan Doyle presents the character in a way that creates a sense of mystery and the grotesque, and prepares us for the dark events that occur later.
First, there is what we learn before we meet Thaddeus Sholto. The pearls he sends anonymously suggest a guilty conscience, and his procedures are furtive. The written messages that accompany the packages are written in variety of different disguised hands, but Holmes can tell the same person is behind them all. The final summons sent to Miss Morstan is written on expensive paper, suggesting wealth. And Holmes—applying his 19th century principles of handwriting analysis—tells Watson that the handwriting suggests the writer is a man of weak character.
This mixture of eccentricity, wealth, and furtiveness is repeated again in the description of Thaddeus's home. When Miss Morstan, Holmes, and Watson finally arrive at his residence, it at first seems a disreputable and sinister place:

"We followed [the servant] down a sordid and common passage, ill lit and worse furnished, until he came to a door upon the right, which he threw open."

Then they encounter Sholto himself, amidst a strange collection of rich furnishings:

"We were all astonished by the appearance of the apartment into which he invited us. In that sorry house it looked as out of place as a diamond of the first water in a setting of brass. The richest and glossiest of curtains and tapestries draped the walls…Two great tiger-skins thrown athwart it increased the suggestion of Eastern luxury....A lamp in the fashion of a silver dove was hung from an almost invisible golden wire in the centre of the room. As it burned it filled the air with a subtle and aromatic odor."

And we are given a glimpse at the unattractive physical appearance and nervous manner of Sholto:

"…[A] small man with a very high head, a bristle of red hair all round the fringe of it, and a bald, shining scalp which shot out from among it like a mountain-peak from fir-trees. He writhed his hands together as he stood, and his features were in a perpetual jerk, now smiling, now scowling, but never for an instant in repose. Nature had given him a pendulous lip, and a too visible line of yellow and irregular teeth, which he strove feebly to conceal by constantly passing his hand over the lower part of his face. In spite of his obtrusive baldness, he gave the impression of youth."

We are also presented with immediate evidence that Sholto is a hypochondriac, and rather self-involved. He asks Dr. Watson to listen to his heart because he has "grave doubts as to my mitral valve." When Watson assures him that his heart sounds normal, he makes this insensitive remark to Miss Morstan:

"Had your father, Miss Morstan, refrained from throwing a strain upon his heart, he might have been alive now."  

This outrages Watson and upsets Miss Morstan. Before Sholto's comment, she hadn't known for certain if her father was dead or alive, and Sholto must have realized this.
So Conan Doyle presents Sholto in an unflattering light, and one that speaks to the moral complexity and dark themes of the novel.