What is Atticus attempting to prove with his line of questioning in To Kill a Mockingbird?

Atticus is trying to prove that Tom Robinson could not physically have attacked Mayella and she wasn’t raped. 

Tom Robinson is a black man accused of raping a white woman.  The evidence is completely circumstantial, meaning that it really doesn’t prove anything definitively.  Atticus questions Bob Ewell, Heck Tate, and Mayella to establish that Mayella never saw a doctor, and to establish what side of the face her injuries were on. 

Atticus was trying to show, it seemed to me, that Mr. Ewell could have beaten up Mayella. … If her right eye was blacked and she was beaten mostly on the right side of the face, it would tend to show that a left-handed person did it.  … But Tom Robinson could easily be left-handed, too. (Ch. 17) 

Atticus then demonstrates that Tom Robinson is crippled.  He has not had full use of his left arm since he injured it in an accident as a young man.  Due to this injury, it would have been very difficult for him to have caused the injuries to Mayella.  Basically, Atticus wants to show that Mayella was never raped, and the injuries were caused by her father. 

“Her father saw it, and the defendant has testified as to his remarks. What did her father do? We don’t know, but there is circumstantial evidence to indicate that Mayella Ewell was beaten savagely by someone who led almost exclusively with his left. …” (Ch. 20) 

Scout believes that Atticus and her brother are counting their chickens, and that this does not prove that Tom Robinson is innocent.  However, Atticus took great pains to point out that there was actually no evidence against Tom Robinson other than the fact that he felt sorry for Mayella and tried to help her.  Unfortunately, for a white jury trying a black man, that was enough.