Lindsay was born in San Antonio, Texas and raised in South Texas. He received his Bachelor of Arts degree in History and Government from St. Mary’s University in 1974, graduating magna cum laude. In 1977, he earned his Juris Doctor from the University of Texas School of Law in Austin. From 1977 to 1979, he was a staff attorney for the Texas Aeronautics Commission. In 1979, he joined the Dallas City Attorney's office, where he progressed through various posts until being named City Attorney in 1992. He was the head of the Federal Litigation Section from 1979 to 1986, a Chief of litigation division/executive assistant city attorney from 1986 to 1990, a First assistant city attorney from 1990 to 1991, an Acting city attorney in 1991, and finally a City Attorney from 1992 to 1998.
On November 8, 1997, Lindsay was nominated by President Bill Clinton to be a United States District Judge of the United States District Court for the Northern District of Texas, to a new seat created by 104 Stat. 5089. He was unanimously confirmed by the United States Senate on March 11, 1998, and received commission on March 17, 1998. Lindsay was the first African American to serve on the federal District Court in Dallas.
On May 21, 2007, Judge Lindsay issued an order blocking enforcement of an ordinance approved by the citizens of Farmers Branch, Texas. The ordinance would have authorized fines against property managers who rented to illegal immigrants.
On January 22, 2015, Judge Lindsay sentenced Barrett Brown for his role in a December 2011 Anonymous hack of Texas-based Stratfor. Brown pleaded guilty to accessory after the fact, interfering with an FBI investigation, and threatening an FBI agent. At the time of sentencing, he had been in jail about 2.5 years. Judge Lindsay sentenced Brown to a total of 63 months in jail and fined him almost $1 million, with the money to be paid to Stratfor and other companies targeted by Anonymous. Jeremy Hammond, who was responsible for the Stratfor breach, pleaded guilty to one count, illegally accessing computer systems. In 2013, Judge Loretta Preska, US district court for the southern district of New York, sentenced Hammond to 10 years, the maximum allowed; news reports make no mention of Preska imposing a fine.
Judge Lindsay is a Trustee of the Center for American and International Law (CAIL). Among other things, CAIL trains law enforcement officers, including FBI personnel.
Lindsay's daughter, Rachel Lindsay, is an associate at Cooper & Scully, PC. She came in third place on season 21 of The Bachelor. On February 13, 2017, ABC announced that she would be the bachelorette on the next season of The Bachelorette, becoming the first African-American lead on either The Bachelor or The Bachelorette.