Charles Darnay’s problem is that he comes from a horrible aristocratic family.
Throughout the course of the novel, Charles Darnay has more than one problem, but they can all be summed up as caused by his family. Charles is now the Marquis St. Evremonde. The family has a history of despicable abuse toward peasants, which also makes them a target for retribution during the French Revolution. Although Darnay himself is innocent, he is guilty by association as far as they are concerned.
Problem: Arrested for treason. Advice: Hire a good lawyer.
The first time his problem is presented to us is when he is on trial in England. He is accused of being a spy, and his brilliant but dysfunctional lawyer Sydney Carton is able to get him acquitted on a technicality. Carton realizes that he and Darnay look a lot alike. He uses that resemblance to create reasonable doubt, and that is that.
Problem: Secretly affiliated with the family that imprisoned your girlfriend’s father. Advice: Tell the truth.
When Darnay becomes involved with Lucie Manette, he has another problem. His fiancĂ© and her father do not know about his past. Essentially he has been lying to them. They think he was just in the wrong place at the wrong time and is an innocent French tutor. This may not be so bad, hiding your name from your fiancĂ© and future in-law, but there is an additional problem. Charles’s family was responsible for imprisoning Lucie’s father, Dr. Manette, in France.
Darnay eventually tells Dr. Manette the truth, but he waits until he is leaving for his honeymoon. By the doctor’s reaction, he knew all along. He stops Charles before he can get into specifics.
"Your confidence in me ought to be returned with full confidence on my part. My present name, though but slightly changed from my mother's, is not, as you will remember, my own. I wish to tell you what that is, and why I am in England."
"Stop!" said the Doctor of Beauvais. (Book 2, Ch. 10)
Dr. Manette does not want to confront the reality of Darnay’s identity, because in doing so he has to confront his past. In fact, the secret’s revelation sends him into a psychotic break of sorts, relapsing to his shoemaking days. After the nine days, he is fine. Lorry arranges for his shoemaking materials to be removed, and that is that. It was good for Darnay to tell him right before leaving, so that his daughter did not see him like that.
Problem: A friend in France needs you, and you cannot go back to France. Advice: Don’t go! It’s a trap.
Alas, this is not advice that Darnay took. Darnay got word that his friend and servant Gabelle had been imprisoned by the revolutionaries. He wanted to go back to France to rescue him. This is extremely dangerous. It is a bad idea all around! Not only does he go, but he brings an entourage with him of basically everyone he knows, including his wife and daughter. Thus he brought them all into danger too. Fortunately, Sydney Carton had another idea. The question then becomes: do you agree to change places with a man, even though it will result in his death?
No, Darnay is too good a man for that. Oh, there is also the fact that he doesn't think that it will work.
"Carton! Dear Carton! It is madness. It cannot be accomplished, it never can be done, it has been attempted, and has always failed. I implore you not to add your death to the bitterness of mine." (Book 3, Ch. 13)
Carton drugs him, and goes to the guillotine in his place. He loves Lucy, and that is all that matters to him. He is willing to die so that she can be happy. It’s romantic. It’s also pretty depressing and creepy. Carton feels that his life is not worth living, and this is the best way to use it.