When analyzing a character, you are analyzing how the author presents that character as a real person. You want to examine how the character speaks, thinks, and acts. You'll also want to gain an understanding for the character's sense of ethics, decide if the character's decisions are wise or unwise, and examine the character's motivations. It can also help a great deal to pay attention to how other characters respond to the character you are analyzing because someone else's attitude towards another person can say a lot. More details for character analysis can be found in the eNotes article titled "How to Write a Character Analysis," and below are a few ideas to help get you started.
Some of the most important characters to analyze are the protagonist Henry Smith, his deceased brother Franklin Smith, and those who accompany Henry while hiking Mount Katahdin: his best friend Sanborn Brigham, his sister Louisa Smith, and the Cambodian immigrant Chay Chouan who was actually guilty of being the driver of the car that killed Franklin. We can tell in the first chapter that Henry is characterized as very quiet, thoughtful, and caring. We can tell he is quiet and thoughtful because very few words are spoken; we are rather introduced to him and his situation through third person accounts of his actions and thoughts. We know he is caring because he is clearly described as being absolutely devoted to his brother, regardless of actually how mean Franklin is in return. Henry is especially in awe of Franklin because he sees Franklin as being better than he is. Henry is small, so he is not as successful at playing rugby or sports in general as Franklin, which makes Henry feel inferior. After Franklin's death, Henry decides to hike Mount Katahdin by himself, partially in honor of Franklin's memory and partially to prove that he is capable of doing it. As the story progresses, Henry learns a great deal about himself, especially the fact that to a fulfilling life means to learn how to deal with the book's theme, "Trouble," and how to love.
Franklin is actually set up as a character foil to Henry; a character foil is a character that has the exact opposite traits of another character, thereby emphasizing those opposite traits. Though Henry worshiped Franklin, as the book progresses, Henry actually learns that Franklin didn't possess the character traits that are worthy of being admired. In contrast to Henry, Franklin was actually self-serving and a bully. We see evidence of the fact that Franklin is a self-serving bully in his treatment of Franklin, specifically, ignoring him and promising to teach him how to climb and take him hiking on Mount Katahdin but then renege on his promise. Before the accident, we learn that Franklin had only given in and agreed to take Henry hiking simply because their father had wanted Franklin to do so.
Hence, through examining both Henry and Franklin, we see that Henry is characterized as thoughtful and caring, while Franklin is characterized as Henry's character foil.