The reader is first introduced to Sydney Carton in a courthose, where he saves Charles Darnay from conviction of treason for the first time. His likeness to Darnay, inspite of his insolent and discheveled demeanor, is unsettling, and Darnay is let off. The contrast between the two (except for the fact that physically they're practically twins, if that makes sense) is remarkable. Darnay eats a good healthy meal and drinks fine wine, Carton sits in a corner and has a bottle of port to himself.
Carton, we soon find, has a drinking problem and many other problems besides. The nature of his sins, besides drink, are never really revealed, but we get the sense that Carton near detests himself and the world. He is apathetic, miserable, and has lost whatever ambitions he had for himself. But there is a sense that he could have been great: he is a more than able lawyer, intelligent, educated. He had an ideal he fell from, and now he feels his life is going nowhere.
A beacon of light enters Sydney's life when Lucie Manette does. She is a beautiful young woman in love with Darnay, and Sydney is in love with her. She is the only onewho eally treats him kindly, and he asks to be treated as a guest in thier home after she and Darnay are married, because he "would die for her and her happiness". She accepts and Carton goes on with his unrequited love.
Both Lucie and Darnay are emigrants from France, but while Lucie is the daughter of a prisoner of the Bastille, Darnay happens to be the descendant of cruel aristocrats. When he is summoned to France to help an old servant, he is captured and charged with treason (on the grounds that he is the Marquis St Evremonde encore, despite the fact that he's lived in England for years and despises the tyranny of his predecesors.) Anyhow, after a long and complicated trial and retrial, Darnay is to be executed.
Upon hearing this, Carton travels to Paris (apparently he speaks like a native), not to comfort Lucie, who loves Darnay with all her heart, not to try to pick her up on the rebound, but to save Darnay.
The only way to do it, he finds, is to switch places with him. To die in his place. And in one of the most moving passages I've ever read, that is what he does. Carton renders himself the hero of the novel in this single, selfless act.