There are several possible themes for this novel, so I am not sure which one you have in mind for your question. I provided a link below that discusses different possible themes, and maybe that can help you out a bit if you don't have one in mind. In general though, to progress a theme, an author uses several different tactics. The first is the plot; they create a setting and plot that furthers their theme. The second is characters; they use the characters to exemplify their theme. The third is the use of symbolism; they create symbols and layers to their novel that can be tied into the theme.
So, let's say that the major theme of this book is prejudice, and how people's prejudices can cause a lot of harm. Harper Lee set the book in the south, in a time period and place where a lot of racism existed. This is the perfect setting to talk about racism and prejudice. Then, she created a plot where that theme would be brought front-and-center. There is the trial of a black man accused of raping a white woman. Racism and prejudice are the key components of such a trial. She also has the plotline of Boo Radley to talk about prejudice on a smaller scale; prejudice against people in society who are different. She then created characters that would promoter her beliefs on prejudice (Atticus, a noble lawyer defending a black man in a hopeless case; Tom, a good black man victimized and killed because of prejudice; Boo, a social recluse who turns out being the hero). Lee also used symbolism with the mockingbird; it represented the innocent Tom Robinson and Boo Radley who were brutally destroyed because of prejudice, just as killing a mockingibird is to destroy an innocent creature.
Through these techniques, Lee promotes her theme of racism and prejudice. All of those techniques can be applied to any theme that you have in mind; good luck!