Charles Dickens had several reasons for writing "A Tale of Two Cities." One reason, and the basic reason any author publishes a novel is to earn money. Dickens had this novel published in a serial format with a few chapters being released each month. The people would get hooked on the story and continue to buy the publication each month so they could see what happened next in the story.
Another reason Dickens wrote this novel was to try something different. He had always written about the Victorian era but with this book he was delving into a new area. This novel would be a romantic novel set in the time of the French Revolution. The two cities being Paris and London. Dickens was not very kind to either city and part of his story spends considerable time discussing the unfairness of the Feudal system and how badly the King treated his subjects.
Most people who have read this book either love it or really hate it. The critics during this era could not agree as to whether or not this experiment in writing was sucessful or not.
I am not sure Dickens wanted to show anything. A novel is not a report or a thesis, or an attempt to demonstrate something. That is not the way great writers work. They try to convey their feelings, their emotions. They are great writers precisely because they are better and have fewer inhibitions than others in exposing their feelings and putting them in words. When successful, they achieve in this way a "truth of the heart" that readers are able to relate to, even after decades or even centuries. The creative process is a mystery, a miracle. In some cases, the trigger is a compelling experience or memory that fires the writer's imagination. Often the writing is slow and painful at the beginning, while the basic building blocks are being devised and the characters are being sketched. Then the novel and the characters may take a life of their own. Dickens was indeed very much affected by his family's experience as well as his own in his early years and was particularly sensitive to the situation fo the poor. This clearly transpires in all his work, including Tale of course. He was also aware that the poor are not necessarily "good" and may indeed become very resentful and violent when humiliated for too long. Maybe when Dickens wrote Tale in 1859, he had a feeling that history had repeated itself: France had experienced another revolution in 1848 and another Napoleon was ruling France when he was writting. This may be why he got interested in the French revolution of 1789. From this perspective, Tale may have been a cautionary tale: if you, the rich, do not pay attention to the plight of the people you may have a bloody revolution on your hands. This warning applied to France, but could very well be extended to England where the conditions of the poor were not much better than in France.