The only characters that take up most of the storyline are the father and the son, and what they are trying to do is survive in the dreary world that has been presented to them. Knowing a lot about the father's past wouldn't change the fact that every single day, he has to get up and survive, find food for himself and his son, fend off barbarians, and search for hope in the hopeless world that they live in. Not revealing a lot about the backstory of the father keeps the reader in that very dreary present; to go back to a better time would be a respite from the difficulty of his trial. By not revealing more, the reader feels the strain, the every day brutality of trying to maintain hope in their world. McCormac is merciless to the reader in this sense; we are thrust, as the father is, into this world where your past doesn't matter. Your personality doesn't matter. Nothing matters except for surviving.
As for the son, there is nothing more to know about him than this life--he was born into it, and has lived it from birth. He has no other memories to dwell on. We learn all we need to know about him through his reactions to strangers, his fear of the barbarians, and his concern from his dad. If we piece together those aspects, his character is full and intriguing.
As far as other characters go, they are not sigficant to the main themes of survival, and a father's love for his son. The mother is no longer in the picture; we get glimpses of her, but not a full image, since dwelling on her is not useful. We are only presented with the father's perspective of other characters that they encounter, once again putting the reader right into his reality.
I hope that helps; good luck!