In this chapter, it seems pretty clear that Atticus does not have any problem with the idea of having capital punishment. However, he does have a problem with how easy it is to get someone sentenced to death. He thinks that capital punishment should not be allowed in cases where the evidence is only circumstantial.
Atticus says that he wants there to be clear proof that someone has committed a capital crime before they can be sentenced to death. He says, for example, that he would want two witnesses to be able to say they saw a person pull a trigger and kill a murder victim. Here's a quote that shows this idea
He said he didn't
have any quarrel with the rape statute, none what ever, but he did
have deep misgivings when the state asked for and the jury gave a
death penalty on purely circumstantial evidence.
So Atticus is okay with the idea of the death penalty but he wants juries to have to have clear proof before they impose it.
The view that Atticus Finch has toward capital punishment is that of an honest and conscientious lawyer:
"The law says 'reasonable doubt,'but I think a defendant's entitled to the shadow of a doubt. There's always the possibility, no matter how improbable, that he's innocent."
As an illustration of Atticus's cogent points, the famous actor, Henry Fonda, produced a movie that is shown often on the classic movie channel; this movie is a film adaptation of a play entitled Twelve Angry Men and the narrative involves the trial of a youth who has apparently stabbed his father to death. Most of the action takes place in the jury room as the jurors first vote 11 to 1 on a guilty charge. But, Henry Fonda's character insists upon closely examining all the evidence and reviewing the testimony of the witnesses, their credibility, etc. for reasonable doubt. Finally, after several votes and more examination and discussion, Henry Fonda's character convinces ten of the others that there is 'reasonable doubt'--the other juror begrudgingly goes along--and the youth is acquitted. Were it not for Fonda's objectively reasonable character, the youth would have been found guilty, just as Tom Robinson's has been.