Great question! There are many potential areas of comparison that you could focus on in comparing these two classics, but if I were you I would examine the way in which both texts present the development of their central characters and show how they mature and develop during the course of the novel.
In Pride and Prejudice, clearly the two central characters who change and mature most are Darcy and Elizabeth. They both, during the course of the novel, learn some very hard truths about themselves and their various failings and come to be better people because of it, learning about their mistakes and then moving to correct them. This also means that they are able to marry. Note for example Lizzie's reaction to Darcy's letter after his first proposal to her, when she has taken time to think about its contents:
Had I been in love, I could not have been more wretchedly blind. But vanity, not love, has been my folly. Pleased with the preference of one, and offended by the neglect of the other, on the very beginning of our acquaintance I have courted prepossession and ignorance, and driven reason away, where either were concerned. Till this moment I never knew myself.
Clearly this shows the way in which self-knowledge leads to the development of character as Lizzie realises just how "blind" and prejudiced she has been and how she has been taken in by first impressions.
Of course, we can identify a similar change in A Tale of Two Cities by focusing on the way in which the character of Syndney Carton is presented as being dynamic. The way in which he begins the novel as a worthless member of the human race who has given his life over to degradation is used to convey the key theme of resurrection in the novel, for he ironically finds his life and truly lives at the moment when he sacrifices it for love. The way in which he does this makes for one of the most moving finales of any novel I have ever read, as he lets himself be cruelly executed so that his true love, Lucie, can escape with her husband and daughter.