Atticus's strategy to defend Tom Robinson is to get Sheriff Tate's testimony from the initial report which demonstrated that Mayella Ewell had been beaten on the right side of her face (In Tate's words, "She was all bunged up on that side of her face.") Next, he cross-examines Bob Ewell, and gets Ewell to write something for him, at which point the judge and jury can plainly see that Ewell is left-handed. A left-handed man would most likely punch a person on the right side of that person's face.
Next, Atticus cross-examines Mayella Ewell, who testifies that she'd asked Tom to bust up a chiffarobe for her, went into the house to get a nickle for him, and "I turned around and 'fore I knew it he was on me. Just run up behind me, he did. He got me round the neck, cussin' me an' sayin' dirt--I fought'n'hollered, but he had me round the neck. He hit me agin an' agin--" (205). Atticus keeps her on the stand until she clarifies her story a bit, then has Robinson stand up to be identified, at which point the whole courtroom can tell that his left arm is shriveled and worthless (Jem says it was caught in a cotton gin and ruined when Robinson was twelve years old).
In short, Atticus's strategy is to clarify that Robinson could not possibly have done what Mayella and Bob Ewell accuse him of. In addition, he offers his own theory during his cross-examination of Mayella: "What did your father see in the window, the crime of rape or the best defense to it?" (213).