TL;DR We cannot be 100% sure of the non existence of Atlas's family, but the reason he use this whole scheme is because Jack is not the only one he need to keep unaware that he is manipulated. Tenenbaum, Steinman, Cohen, Ryan and other would had used him to their own ends if they were aware they could do so. Specialty Tenenbaum since she was part of the experiment that made Jack who he is. This might also reveal Atlas real identity, which is still secret.
As for the identity of Atlas's family, he say the name of his wife and son, it's Moira and Patrick.
My wife, Moira—she's a right pain in the neck. But she's a beauty and she means the world to me. I can't help but feel God's punishing me for bringing her and Patrick to this place. I thought this would be a better life for us. Can you imagine a bigger fool than that?
You may have noticed that those names are also used in a stage production in the Fleet Hall, Patrick and Moira.
So the whole existence of those characters is doubtful.
And within the end, near the start of the Proving Grounds level, Fontaine say he don't have a family :
I really wound you up with that wife and child bit: "Oh, me poor Moira. Ah, me wee baby Patrick." Maybe one day I'll get me a real family. They play well with the suckers.
But you could also note than (small spoiler for Bioshok 2)
From Frank Fontaine wikia article, BioShock 2 section :
"In the office there is also a large portrait on the wall of a man who resembles Fontaine standing next to a woman and a small boy. The picture is entitled "The Fontaine Family," but it is unclear who the people depicted in it might be, since Fontaine told Jack that he never had a wife or child."
(you cloud see the legend here)
Those might be Frank Fontaine with Moira and Patrick, or a young Frank Fontaine with his mother and father.
As for Fontaine's motivation, there what the Patrick and Moira Wikia article say about it :
Even though Fontaine had Jack completely under his control, he seemed to enjoy getting Jack to believe that Ryan is evil by conning him. It was not necessary for Fontaine to do this, as a simple command to kill Ryan would have worked just as well, but giving Jack the impression that Ryan would murder someone's wife and child must have been a tactic to make Fontaine's commands less overt.
The other point is that Jack is not the only one who need to be unaware of his real identity and that he is manipulated. Tenenbaum would had forced him to save the little sister instead of harvesting them or Steinman would had asked him to kindly lie down on the surgery table.
Also, revealing the identity of Jack as the subject of the experiment on mind control imprinting may also have reveal the real identity of Atlas as Fontaine to some who still ignore it. Especially Tenenbaum, who purchased Jack's embryo on behalf of Fontaine and helped raising him.