Chief Justice Chew was born in El Paso, Texas. He is the first son of Wellington Yee Chew, a prominent attorney and community leader, and Patricia Mary Chew, an elementary school teacher and principal, both deceased.

Raised in Northeast El Paso and graduated from Irvin High School in 1966, he attended the University of Texas at El Paso for one year before accepting congressional appointment of Congressman Richard C. White to the United States Naval Academy. He graduated from the Academy with a Bachelor of Science in 1971 and was commissioned as an Ensign in the United States Navy.

His first assignment was on the U.S.S. Claud Jones (DE-1033), homeported in Pearl Harbor, Hawaii as Communications Officer. In early 1972 he was fleeted up to Operations Officer, and, in that capacity, he participated in a WestPac deployment during which the Claud Jones was tasked as the principal interdiction/notification ship, under the direct operational command of Commander, Seventh Fleet, following the mining of Hai Phong harbor. Justice Chew was awarded the Navy Achievement Medal following the completion of that deployment. He also participated in four extended special operations deployments made by the Claud Jones before the ship was decommissioned and transferred to the Indonesian navy in 1974.

Then Lieutenant Chew was selected for assignment as the Flag Lieutenant/Personal Aide to Rear Admiral Justin Langille, III, Commander, Cruiser Destroyer Group Twelve, homeported in Mayport, Florida in late 1974. As the Flag Lieutenant, he was embarked with the admirals staff on numerous ships including the aircraft carriers John F. Kennedy, Forrestal, and Independence and cruisers/destroyers Leahy. In addition to one deployment to the Mediterranean, he also participated in the exchange visit of the U.S.S. Leahy to Leningrad, USSR commemorating the 30th anniversary of the end of World War II.

Chief Justice Chew, after leaving active duty, served in the U.S. Naval Reserve and commanded several naval reserve units. He attained the rank of Lieutenant Commander.

Chief Justice Chew graduated from the Southern Methodist University School of Law in 1978 and returned to El Paso to enter into the private practice of law. He entered into partnership with Paul M. Douglass, who practiced with Justice Chew's father. Though they maintained a general law practice, both partners were board certified specialists in Immigration and Nationality Law. The firm expanded when Justice Chew's sister, Linda Yee Chew, joined the firm in 1985. He was also listed in the 1994/95 and 1995/96 editions of The Best Lawyers in America. In 2002, the Asian Pacific Islander Section of the State Bar of Texas established the "Justice David Wellington Chew Award" to honor an outstanding Texas Asian American lawyer each year. 

In 1989, he was elected to and served one term as the West-Central City Representative to the El Paso City Council.

In 1994 Democratic primary and General Election, he was elected to the Eighth District Court of Appeals and sworn in on January 1, 1995. Since taking the bench and through December 31, 2006, he has authored over 1,200 opinions on the merits. In April 1999, he served on the Texas Supreme Court on two cases on the commission of Governor George W. Bush.  In October 2006, he was appointed Chief Justice by Governor Rick Rick Perry.  Chief Justice Chew is the first Asian American to serve as a chief justice in Texas and he is the highest ranking Asian American elected official in Texas.