I tried to come up with a sequence of events to explain their story!
I'm not entirely sure if the Aftons are later on turned into vampires or born that way, could be a variation or sorts.
Everything is good in the Afton Family.
Mrs. Afton passes away/is killed. William Afton loses his mind over it.
He conducts a series of experiments involving the creation of flesh + machinery amalgamates. He wants to fabricate the perfect vessel-body for his wife once she's somehow ressurected.
Children around the town go missing, and their blood is used in the experiments.
William begins adding metal pieces to himself in order to become further more invincible. His mind is slowly deteriorating as the years go by.
Crying Child suffers from a fatal head injury in an accident. William rushes to bring him back to life and is "successful". That only drives him forward to continue with the experiments.
Crying Child becomes an insatiable beast-like amalgamate fused with his favorite plush toy.
Mike, shocked by his father's creation and traumatized by his mother's passing, runs away from home and swears to never drink from anyone's blood.
Elizabeth welcomes her little brother's new appearance and habors resentment for Mike. She's awaiting eagerly to become a machine and make her dad proud.
William allows the amalgamates to run freely around their territory and kill anyone that might come near.
One of William's amalgamates "Ennard" escaped its confinement and found a weakened Mike. The two agree to fuse together and stop William's plans.
William turns himself into a machine named "Springtrap".
With much difficulty, Mike returns home and is welcomed by Springtrap. The father explains his motives and attempts to convince his son to take part in the experiments in order to "be part of the family again". Mike denies the offer, and the two fight.
Too weak to keep going without blood, Mike is defeated. He is confined, and forced to add metal parts to himself in order to become stronger.
With much sorrow and anger, Mike swears to put an end to his family's bloodline. He is helped by the missing children's souls.
It is unclear whether or not Ennard is the specific vessel-body designed for Mrs. Afton and if it is possessed by her soul.