"For us, it's a case. It's a cause. It's a hashtag," Crump told Reuters. "For them ... it's their family. This is their blood."
Floyd, a 46-year-old African American, died after Chauvin, who is white, kneeled on his neck for more than nine minutes. His death, captured on bystander video, sparked worldwide protests over racism and police brutality.
Chauvin, who faces up to 40 years in prison on murder and manslaughter charges, has pleaded not guilty.
The case is familiar terrain for Crump, who is frequently called upon to represent the families of slain African Americans in civil lawsuits, including Trayvon Martin, a teen shot dead in 2013 by a neighborhood watchman, and Breonna Taylor, who died during a botched police raid.
Crump, 51, who grew up in rural North Carolina and attended segregated schools for most of elementary school, sees his role as a civil rights advocate who keeps media attention on Black victims who otherwise might not receive "full justice" under the U.S. Constitution.
Grand juries rarely indict police officers for killing a suspect in the line of duty in the United States, particularly when the victim is Black, according to legal experts.
"What we're doing is continuing to make the arguments in the court of public opinion," said Crump. "The court of law is not very kind to marginalized minorities."
With that in mind, Crump often resorts to civil litigation.
It was Crump who helped the Floyd family sue the city of Minneapolis, resulting in a $27 million settlement that he has called the largest pre-trial settlement of a wrongful death lawsuit in U.S. history.
The settlement, coming two weeks before the trial opened, came under criticism for its potential influence on jurors being selected for the criminal trial, including from the judge who called it "unfortunate."
Crump dismissed the criticism, saying that white families frequently receive civil settlements before criminal "justice" in similar cases.