Looking at the big picture, we might be tempted to say that Atticus fails to achieve justice. Despite his brilliant defense of Tom Robinson, Tom is still convicted by the jury. Atticus would like to appeal the case, but Tom (understandably) falls into despair, makes a suicidal escape attempt from the jail yard, and is shot dead. Tom's family are left without a provider. Atticus' son Jem is disappointed and disillusioned. Most of the white people in the town (e.g. Aunt Alexandra's tea ladies) go on in their cluelessness, baffled and irritated that the black community are so upset. So, to all appearances, it is game, set, and match for the Devil.
However, as Miss Maudie points out to Jem in Chapter 22,
"I thought, Atticus Finch won't win, he can't win, but he's the only man in these parts who can keep a jury out so long in a case like that. And I thought to myself, well, we're making a step - it's just a baby-step, but it's a step."
Atticus was unable to get true justice for Tom, but through his examination of Tom, his cross-examinations of Bob and Mayella Ewell, and his closing statement, he made it completely clear to everyone in Maycomb what really happened in the Ewells' cabin. Although the jury finally did the monstrous injustice of convicting Tom, Atticus had made it very difficult for them to do so, and that is clear by how long they deliberated.
In a sense, Atticus cleared Tom's name, and everyone in Maycomb knows it. The black community know it, and they shower gifts of food on Atticus the next day. Bob Ewell knows it, and he also knows that Atticus has damaged his and Mayella's credibility forever. That is why, despite having won the case, Bob Ewell keeps trying to get revenge on Atticus.
Atticus does many other things, large and small, that achieve justice in small ways. He does pro bono legal work for those in the community who need it. He stands up to his family when they say snobbish or racist things. He works hard, and though a good shot, is almost a pacifist.
But the other major way that Atticus achieves justice is in the way he raises his children. Many pages could be written about the ways in which Atticus strives to pass on his values to his children - by living with integrity, by talking to them about it, by the questions he asks them and by the way he answers their questions. He is not brainwashing, but training. He respects their minds. By the end of the novel it is clear that he is in the process of achieving justice by releasing on the world two intelligent, moral, courageous and compassionate human beings.