Charles wants to give up his title in Chapter 9. He is the nephew of the Marquis. He tells his uncle that the family name, Darnay, stands for "fear and slavery" and he wants to disassociate himself from it. He tells his uncle that his family has treated every human creature that interfered with the family's desires and pleasures with cruelty. The heartless Marquis, however, pooh-poohs Charles' protests and tells him he cannot renounce his title, it is his destiny. The Marquis is later found dead.
This section shows how Charles Darnay is not a typical French aristocrat - one who callously can run over a child with his carriage and throw coins to the father, or one who, like the Marquis, can cruelly pass a woman who asks for a mere stone marker for her husband's grave. Although Charles is an aristocrat, he does not have an aristocrat's pride and haughty attitude towards the poor. This is why the French Revolution was so bloody -- the titled French, as well as the monarchs (Louis XVI when the revolution broke out in 1789). The poor were not only starving for centuries, but they were treated with total disdain by the wealthy, and Charles Darnay wants no part of this. He is more Republican than aristocrat.
You can read this exchange here on eNotes at the link below.
In this section of A Tale of Two Cities, Charles doesn't actually renounce his title and inheritance, but he certainly makes his intentions to do so quite clear to his uncle--the same uncle who probably tried to get him convicted of treason and who would just as soon Charles was dead.
This intent to renunciate is based on Charles' good character as well as his understanding that his father and his uncle created a tainted legacy, one which he had no desire to inherit. In chapter 9 he says:
What is it but a wilderness of misery and ruin?...To the eye it is fair enough here; but seen in its integrity, under the sky, and by the daylight, it is a crumbling tower of waste, mismanagement, extortion, debt, mortgage, hunger, nakedness, and suffering.
Clearly Charles is aware of the inequality and inherent unfairness of an aristocratic system,as well as the fact that his family holdings are steeped in evil and pain--even though he is unaware of the exact history. This intent to denunciate not only makes him a good man but a wise one. Now, if only he had actually done it....