Why didn't Atticus mention his shooting ability to his kids?

Atticus is a moral, compassionate, humble man, who wants his actions to reflect his tolerant beliefs. In chapter 10, Atticus is hesitant to shoot Tim Johnson with Sheriff Tate's gun for several reasons. Miss Maudie tells Jem and Scout that Atticus is such a talented marksman he feels his ability to shoot accurately is unfair to the game he targets. Maudie also tells them that Atticus has a "civilized heart" and that he believes his marksmanship is a gift from God. Apparently, Atticus had decided that he would not shoot anything unless he absolutely had to.

When Jem asks why Atticus never bragged about his marksmanship, Maudie proceeds to tell the children, "People in their right minds never take pride in their talents" (Lee, 102). Essentially, Atticus is too humble to take pride in his God-given ability. In the next chapter, Atticus makes Jem read to Mrs. Dubose, who is dying from a chronic disease, and explains to his children,

"I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand" (Lee, 115).

Atticus's comment gives additional insight into why he would not brag about being an expert marksman. He does not want to raise his children thinking that violence is the answer to anything. Atticus would prefer that his children admire him for his empathy, wisdom, and civility, instead of his marksmanship. 

Why didn't Atticus in To Kill a Mockingbird tell his kids about his "one-shot Finch" past?

Atticus Finch is not only the father of the young protagonist of Harper Lee's novel To Kill a Mockingbird; he is also the novel's moral conscious. An infinitely decent and wise lawyer dedicated to the pursuit of justice, he is raising his children with the same values that shaped his character. In addition, Atticus, Scout emphasizes in Chapter 10, is relatively old to be the father of young children. He appears to Jem and Scout as "feeble" and unable to parent the way younger couples are seen parenting. As if to further emphasize the point, Scout goes on to say:

"Our father didn't do anything. He worked in an office, not in a drugstore. Atticus did not drive a dump-truck for the county, he was not the sheriff, he did not farm, work in a garage, or do anything that could possibly arouse the admiration of anyone."

Scout is six years old and Jem is ten when To Kill a Mockingbird begins. They are too young to understand much that occurs in the world, existing as they do in the small town of Maycomb. To children of such age, a profession like lawyer is hardly the stuff of which dreams are made. Finally, Atticus eschews the use of firearms. Scout notes in this chapter that when Atticus presented his children with air-rifles, he refused to have anything to do with the guns, including teaching his children how to use them. Scout quotes Uncle Jack as asserting that "Atticus wasn't interested in guns."

Atticus's aversion to guns assumes considerably greater significance later in Chapter 10 when Maycomb County Sheriff Heck Tate enlists the lawyer's assistance in shooting a rabid dog. When Atticus, with much prompting by the sheriff, takes the rifle, aims and fires, killing the dog with one shot, Scout and Jem are astonished to discover their feeble, old father is actually an expert shot. The answer to the question--why did not Atticus ever tell his children about his prowess with a rifle--lies, perhaps, in Miss Maudie's discussion with Scout and Jem:

"If your father's anything, he's civilized in his heart. Marksmanship's a gift of God, a talent.... I think maybe he put his gun down when he realized that God had given him an unfair advantage over most living things. I guess he decided he wouldn't shoot till he had to, and he had to today."

Atticus never discussed his skills as a marksman because he was an inherently modest and decent human being who didn't like killing anything. When he presented the air-rifles to Jem and Scout, it was with the proviso that they never shoot at any living thing. A realist, however, he recognizes that, being children, they will be tempted to shoot at birds, providing the novel's central theme: "shoot all the bluejays you want...but remember it's a sin to kill a mockingbird." Atticus never told his children about this particular skill because, besides being modest, he takes no real pride in it.