Carton throughout A Tale of Two Cities is shown as a man who hates who he is, and wishes he could be someone different but cannot. I think one of the best quotes to represent this is in "Fellow of No Delicacy", where he says to Lucie,
"I have had unformed ideas of striving afresh, beginning anew, shaking off sloth and sensuality, and fighting out the abandoned fight. A dream, all a dream, that ends in nothing, and leaves the sleeper where he lay down, but I wish you to know that you inspired it.”
He only truly feels he has done something good in his life when he dies in the place of Darnay, for the sake of Lucie and her family.
But... why? I know Dickens never really writes in A Tale of Two Cities about Carton's past, but are there other clues in the text explaining why he feels unredeemable?