Authorities say Sandra Bland hanged herself, but her family says she would not have taken her own life.
A video was found on her Facebook page in which she said she was suffering from "a little bit of depression" and post-traumatic stress disorder.
Her death follows a series of cases in which black people have died in police custody or during arrest.
Bland had been stopped by police after improperly signalling a lane change while driving. She was arrested after allegedly kicking a police officer after the traffic stop.
Family members were shocked by her death, unable to believe she would have killed herself.
A second video showed Bland saying "Depression is nothing but the devil" and that "it's a daily struggle".
The FBI has launched an investigation, following the start of a separate inquiry by the Texas Rangers law enforcement agency.
The 28-year-old had just moved to Texas from Illinois to start a new job.
Her sister Shante Needham said Bland called her from jail, saying she did not know why she had been arrested and that an officer had possibly broken her arm.
Bland died by asphyxiation using a plastic bag to hang herself in her cell, according to Waller County District Attorney Elton Mathias.
Video footage did not show what happened inside Bland's cell, but did suggest no one entered or left it until someone found her unconscious.
The Texas Commission on Jail Standards cited the Waller County Jail three years ago for not checking all inmates once an hour. Another inmate hanged himself with a bed sheet in 2012.
Jail Sheriff Glen Smith said his staff checked on Bland less than an hour before she was found dead.
Texas state senator Royce West asked the Texas Department of Public Safety to release video of the arrest and called her death "suspicious".
The FBI is to investigate the case of a black woman found dead in a Texas jail cell, apparently from a suicide.