Sydney Carton's choice of employment stems from his personality and the life that he has led. Dickens doesn't give us much background at all as to his past life, but whatever is was, it has led him to his current position where he drinks heavily and is "a disappointing drudge... [who cares] for no man on earth, and no man cares for [him]." He has no ambition to do anything, so he gets by as best as possible. He also describes himself as a man who has always done things for others and never for himself (Chapter-The Jackal).
When he speaks with Lucy in the chapter titled "The Fellow of No Delicacy", he tells her that all his hope in life has died. He had a glimmer of hope upon meeting her, but he realizes he is not good enough for her, so instead he confesses his love for her and then asks her to keep it secret forever and to just think about him tenderly every once in a while.
Carton has a phlegmatic personality that won't allow him to pass beyond his sullen and drunken state. Ironically, he is the one who in a spurt of ambition and heroism, saves Lucy, Darnay, and their family in the end. Throughout the novel he serves as the alter-ego for Darnay. For more on his charcter, check out the link below
Sydney Carton is in his forties by the time we meet him in the book, and we know nothing about him up to this point. His brilliant legal mind is overshadowed by his alcoholism. He has no purpose or goals in life and has become a cynic who describes himself as "a dissolute dog who has never done any good and never will." He helps Stryver win cases and climb the ladder to success, refusing to go on his own. Sydney says he's "incapable of all the higher and better flights of men", indicating the loss of his self-esteem.
The only way Sydney feels he can redeem himself is by giving Lucie the gift of her family. His secret love for her convinces him to finally find meaning in his life by taking Darnay's place on the guillotine.