Redmond was born in Chicago. He graduated from Lincoln Christian College and Seminary in 1979 and was ordained as a minister. Redmond later attended Murray State University, earning a degree in special education. He served in the United States Army Reserve from 1980 until 1983 as part of an Army chaplain candidate program. Redmond was a minister for the Santa Fe Christian Church and a teacher at University of New Mexico-Los Alamos.
He ran for Congress in 1996 and was badly defeated by the district's longtime Democratic incumbent, Bill Richardson. However, only three months after that contest, Richardson resigned to become United States Ambassador to the United Nations. Redmond entered the special election for the balance of Richardson's term. He was initially a huge underdog even though the Democratic candidate, state Insurance Commissioner Eric Serna, was deeply unpopular with voters in the district. However, a Green candidate siphoned off enough votes from Serna to allow Redmond to win by only 3,000 votes.
Despite representing a heavily Democratic district, Redmond had an unshakably conservative voting record. He ran for a full term in 1998 and was soundly defeated by state Attorney General Tom Udall, who received 53% over Redmond's 43%. He became the Republican nominee for the U.S. Senate in 2000, but was defeated by incumbent Jeff Bingaman 61% to Redmond's 38%.