How is the theme of duality evident in A Tale of Two Cities by Charles Dickens?

The theme of duality in A Tale of Two Cities is best seen in Dickens's respective portrayals of England and Revolutionary France. England is presented as a place of stability and peace, where good order and the rule of law prevail. Revolutionary France, on the other hand, is mired in chaos, bloodshed, and constant upheaval.
That's not to say that Dickens is a thoroughgoing jingoist; the trumped-up charge of treason leveled against Charles Darnay hardly redounds to the credit of the English criminal justice system. But even here it's notable that Darnay still ends up being acquitted, something that doesn't happen—and indeed could never happen—when he's hauled up before a Revolutionary tribunal later on in the story. It's notable too that an Englishman, Sydney Carton, ends up saving the life of a Frenchman, Charles Darnay. In the dualistic world-view presented here, one could see this as symbolic of Dickens's belief that English values offer the French a way out of their general malaise.

In A Tale of Two Cities, the theme of duality is most present in both the contrast and comparison between Paris and London, and Charles Darnay and Sydney Carton.
Paris is presented as a hotbed of chaos and violence. Savagery reigns and blood runs through the streets beneath the guillotine. This is contrasted with the ordered, more peaceful London, where the social order is presented as in balance.
Charles and Sydney are two men who physically resemble one another, but are polar opposites in manner. Charles is kind and a gentleman, while Sydney is a self-loathing alcoholic. Charles is loved by many people, while Sydney inspires pity at best. In a strange way, these two men are halves of the same individual. Sydney cannot be whole until he becomes more like Darnay by becoming a hero, dying to save Charles and the happiness of Lucie Manette.

A duality in literature refers to two opposing parts of the same whole. For example, good and evil are the two opposing parts of human beings. There are several dualities in Dickens' A Tale of Two Cities. 
The book starts by presenting a series of dualities:

"It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the season of light, it was the season of darkness, it was the spring of hope, it was the winter of despair" (page 3).

These dualities are meant to represent the duality of the French Revolution, which is both hopeful in its intent to curb the excesses of the French monarchy, yet it also brings despair with its turn to excessive bloodshed. 
At the beginning of the book, there is an explicit contrast between the duality of England and France. Dickens writes, "Spiritual revelations were conceded to England at that favored period" (page 3). Later, Dickens writes that "France [was] less favored on the whole as to matters spiritual than her sister of the shield and trident" (page 3). As France becomes embroiled in the chaos and bloodshed of the French Revolution, England remains the land of peace and a refuge from France. Charles Darnay must leave France and live in England to escape from his family's evil reputation as French aristocrats. 
Similarly, Sydney Carton and Charles Darnay are also presented as dualities. They are "so like each other in feature, so unlike each other in manner" (page 54). They look so similar as to be mistaken for each other, but they are different  in that Carton is disliked and disreputable, while Darnay is respected and adored (particularly by Lucie Manette). In the end, when Carton saves Darnay, Carton becomes good, and the two sides of this duality are reconciled.