Less than 12 hours earlier, a 911 call led paramedics to find the father dead in the yard. Inside the home they found the other three victims, all shot to death.
The youngest victim, Dickson's 14-year-old brother, was apparently trying to hide behind a clothes dryer when he slain, authorities said.
Investigators have not determined a motive for the killings, Griffin said.
Griffin said Dickson does not have a lawyer. She would not say if he is cooperating with investigators.
The coroner's office identified the victims as Dickson's 46-year-old father, Samuel Andrew Dickson Jr.; his 46-year-old stepmother, Martiza Hurtado Dickson; his 19-year-old stepsister, Melissa Giliam Salazar; and his 14-year-old brother, Taylor Alex Dickson.
Neighbor Melissa Funk said Nathaniel Dickson moved out of his family's home after graduating from high school last year, but had moved back in the past few weeks. She said the teen played video games with her 16-year-old son and never appeared to cause any problems.
Authorities were called to the scene at about 9:30 a.m. by a 911 caller who reported someone had possibly fallen from a car.
Fred Gilbert, who has lived in the neighborhood for about 20 years, said most of the houses along the quiet, dead-end street near Easley are on one or more acre tracts.
"I've never had any problems, and I bought this house from my parents," said Gilbert, who lives about a quarter-mile from where the bodies were found.
Located in northwestern South Carolina, Easley is a city of about 19,000 people some 15 miles west of Greenville.