Since Atticus is a single parent and hard-working, his relationship with his children is a bit unusual. He clearly does care about them, and is concerned with their moral upbringing, but is mostly a hands-off parent.
Scout, the narrator, introduces him like this.
Jem and I found our father satisfactory: he played with us, read to us, and treated us with courteous detachment. (ch 1)
There are several examples of this “courteous detachment” and Atticus’s rather hands-off child-rearing tactics. The children decide that he is old, and does not do anything exciting.
Atticus was never too tired to play keep-away, but when Jem wanted to tackle him Atticus would say, "I'm too old for that, son." (ch 10)
When Atticus is able to shoot the rabid dog, the children realize there is a whole side to their father they have never seen. They are quite interested, but Atticus shrugs it off.
Although Atticus forces his children to mind Calpurnia, the cook and housekeeper, and gives them moral lectures, he admits to his brother that he only threatens to punish them and never actually does.
So far I've been able to get by with threats. Jack, she minds me as well as she can. Doesn't come up to scratch half the time, but she tries. (ch 9)
Atticus believes that all people, even children, should be treated with respect. He does not talk down to his children and does not lie to them. He is always honest and straightforward, and listens to both sides, and they respect him for that.
Atticus makes sure his children are growing up to be good people. He is always willing to talk.
Despite the absence of a mother, Jem and Scout have been raised to regard women as equals. (enotes themes)
Times were different back then. For a single father to raise a boy and a girl would have been difficult. Considering Atticus’s responsibilities and the moral fiber of his children, he did a good job raising them.
The relationship between the children and their father is a close one. Atticus genuinely loves and cares for them and attempts to instil his own values into them. However, he doesn't preach to them, in fact he never talks down to them at all. Sometimes the children don't really understand him, and even feel ashamed of him for being old (from their point of view), and for not doing the kind of things that other children's fathers do, like playing sports. However, their respect for him increases mightily when they realise that he is a deadly shot with a gun, as revealed in the incident of the mad dog. The most important point, however, is that they come to respect him for his moral courage and integrity, as demonstrated most notably during Tom Robinson's trial.