Bradley grew up in a single parent household, and learned the value of hard work from his mother. He attended Cheyney State College, graduating in 1964 with a degree in Education. His first job was teaching sixth grade. While he was teaching, he moonlighted at WDAS in Philadelphia, working for free and later, minimum wage. He programmed music, read news, and covered basketball games.

His introduction to news reporting came during the riots in Philadelphia in the 1960s. In 1967, he landed a full-time job at the CBS-owned New York radio station WCBS in 1967.

In 1971, he moved to Paris, France and was living off of savings. As he ran out of money, he became a stringer for CBS News. He covered the Paris Peace Talks.

In 1972, he was transferred to Saigon, to cover the Vietnam War. He spent time in Phnom Penh covering the war in Cambodia. While covering the war, he was injured by a mortar round. He had shrapnel wounds to his back and arm.

In 1974, he moved to Washington, DC. He covered the Carter campaign in 1976 and then became CBS News' White House correspondent until 1978. From 1978 to 1981, he served as principal correspondent for CBS Reports.

In 1981, he joined the staff of 60 Minutes, when Dan Rather left to replace Walter Cronkite as the anchor of the CBS Evening News. He has been there for over 20 years, and is the only male reporter on the program with an earring.

He is married to the artist Patricia Blanchett. He has homes in Woody Creek, Colorado and New York, New York.