Why is Madame Defarge so merciless towards Charles and his family?

Madame Defarge sees Charles and his family as aristocratic. She has a vendetta for anyone that has any connection with the high class regardless of the situation. She does not care that Charles renounced his titles. She only wants to see people she deems are not "common" die.

Her hatred for Charles stems deep within the main plot of the story, the one which sent Dr. Mannette behind bars in the first place.

The father and uncle of Charles Evremondes, twin brothers, had wronged and killed Madame Defarge's family. She is the little girl that dying boy had hidden somewhere far away.

She lost her entire family to the entitlement of these "Noble Aristocrats". This plants the seed of hatred in heart for the upper class, and Charles being the direct inheritor of the Evremondes, was the center for her hatred. This is the reason why she wanted Charles and his family to die, no matter what.