The Afton family is the main focus of the FNaF storyline. It is currently unconfirmed how many people there are in this family, but it is speculated to be either 5 or 6 members.
William Afton - better known as his main alias 'Purple Guy' - was the CEO of Afton Robotics, the designer of the Funtime Animatronics, and the serial killer who murdered numerous children, who went on to become the animatronic ghosts.
He eventually discovered the ghosts of his victims were alive and dismantled their 'suits'. They chased him, however, and he took shelter in his trusty Golden Bonnie Suit. The suit - due to his haste to get into it, and the moisture of the room- malfunctioned, and killed him. He was locked inside the room he died in, and eventually, when FNaF 3 came around, he reemerged to hunt down his son Michael, who was the night guard at the time.
When Michael set the place on fire on the 'fifth night', he survived. He was later drawn to the FNaF 6 location, where he was trapped in the vents with Molten Freddy, Scrap Baby and Lefty (the puppet), while Henry set the place on fire, this time killing him. Michael also died in the fire.
Not much is known about William's wife. It is speculated, with evidence, that after the death of her daughter Elizabeth, she either left William or died, and William preserved her memory inside the animatronic Ballora. This is all speculation, however.
Following the timeline, we first see Michael in FNaF 4, as the Crying Child's older brother, who bullied him to the point of tears. His bullying went too far, however, when he shoved his younger brother into Fredbear's mouth, and his younger brother is fatally injured, never recovering and going into a coma for 7 days before dying.
We next see him in FNaF Sister Location, where he poses as the technician 'Eggs Benedict' and sets about exploring Circus Baby's Pizza World in search of his sister Elizabeth, who possessed Circus Baby. Whether he was aware or not of his sister's evil plans is unknown, but at the end of the fifth night - if following the game correctly - he is scooped and Ennard - Ballora, Funtime Freddy, Funtime Foxy and baby merged together - takes its place inside his body, before his body eventually crumbles under the weight of having them all inside him five days later. They leave his body to avoid suspicion, but Michael survives due to the Scooper's technology.
Next, he appears in FNaF 1, posing as the Night Guard Mike Schmidt. Then he appears in FNaF 2 as Fritz Smith, and then FNaF 3, where he battles his father - who is now Springtrap - and frees the spirits of Freddy, Bonnie, Chica, Foxy and Golden Freddy (his brother). He sets Fazbear Fright on fire, unaware that his father is still alive.
However, sometime between FNaF 3 and FNaF: Pizza Simulator, he is contacted by Henry, who is under the guise as Cassette Man, and employed once again by Fredbear Entertainment, where his job is to gather the animatronics that falls under paragraph four by the end of 'night 6'. Paragraph four being animatronics who still hold Remnants (ghosts/spirits). Henry locks them all inside the vent system, Michael included, and sets the place alight.
Elizabeth is the middle child of the Afton family, who dies at the hands of Circus Baby, who she eventually possesses. Circus Baby twists her mind, making her a dark and evil killer. She escapes Circus Baby's Pizza World by taking control of her brother's body, but between FNaF SL and FNaF 6, when Molten Freddy is formed, she is kicked out of the group and returns to the FNaF SL location where she becomes Scrap Baby. She is drawn to her brother in FNaF 6, and salvaged by him, before being trapped in the vents and burned- being put to rest.
The FNaF 4 Crying Child appears in all games leading up to FNaF 4 as Golden Freddy. In FNaF SL, he witnesses (not on screen) his sister being grabbed/killed by Baby, and it leads to his horrifying fear of animatronics. His father becomes distant, leaving his older brother Michael to look after him. Michael, however, bullies him relentlessly, and when his pranks go too far, Crying Child gets his head stuck in Golden Freddy's mouth, and gets chomped. He is left in a coma for 7 days, where he plays through nightmares (the FNaF 4 main gameplay) and eventually succumbs to his injuries.
He goes on to possess Golden Freddy, whether being put there by the Puppet or not is unknown, and he reunites with his brother in FNaF 1, where he tries to reach out to him through hallucinations. In FNaF 2, he turns hostile and attacks his brother with hallucinations- resulting in game crashes. In FNaF 3, he is finally put to rest in the Happiest Day Minigame.