Characterization is accomplished in literature in two ways: directly and indirectly. Direct characterization is when the author simply provides direct description about a who a character is. A made-up example might be something like, "He was an ugly and evil villain who would stop at nothing to get revenge on his foes."
Most often, author's are not so obvious nor direct. Indirect characterization is used when characters are described through one of the following avenues: appearance, thoughts, actions, speech, and/or the reactions of others to them. Consider the first description of Miss Manette in chapter four (and remember, the reader is seeing her for the first time through Mr. Lorry's point of view):
...a young lady of not more than seventeen...a short, slight, pretty figure, a quantity of golden hair, a pair of blue eyes that met his own with an inquiring look, and a forehead with a signgular capacity...an expression that was not quite one of perplexity, or wonder, or alarm, or merely of a bright fixed attention, though it included all the four expressions...
In this very opening description the reader is given a full, but not complete, picture of Lucie Manette (Darnay). As her character unfolds throughout the story, she is revealed to be a passive character who, though somewhat frail and certainly not dominant, has such a capacity for affection that others are changed by her.
The process of characterization (or character analysis) begins with collecting evidence (in the form of quotes) of both direct and indirect characterization provided by the author. In addition to this, note any significant changes the character goes through in the course of the story. Once you have collected several pieces of evidence and noted change, you can use your own words to describe and ultimately define that character. A character who does not change is considered static while a dynamic character is one who does change. In your analysis, be sure to show evidence from the beginning, middle, and end of the text for completeness.