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