Almond was born in Charlottesville and raised in Orange County, Virginia. Almond attended Virginia Tech and served as a private in the Students Army Training Corps in 1917 and 1918 in World War I, after which he taught school in Locust Grove, Virginia. He served as a high school principal, and earned an LL.B. from the University of Virginia School of Law in 1923. He married Josephine Katherine Minter in 1925. Almond was a state court judge between 1933 and 1945. He was elected to the U.S. House of Representatives, serving in the 79th and 80th Congresses.
Almond resigned his Congressional seat in 1948, when he was elected Virginia's attorney general. He argued the state's case for segregation of public schools before the Supreme Court in the case of Davis v. County School Board of Prince Edward County, which was consolidated with Brown v. Board of Education (). In 1957, he was elected governor, and took office in January 1958 for a single term that ended in 1962. He succeeded Governor Thomas B. Stanley. His major accomplishment as Governor was ending massive resistance against the desegregation of schools, in opposition to other high-profile southern politicians, such as Virginia Senator Harry F. Byrd and Arkansas Governor Orval Faubus. Almond realized that opposition to desegregation was ultimately futile as the state continued to lose in the courts; when Virginia's massive resistance laws were declared unconstitutional he changed the state's policy and thereby earned the wrath of the Byrd Organization.
Almond had campaigned for President John F. Kennedy, who nominated him to be a judge on the United States Court of Customs and Patent Appeals (CCPA). Senator Byrd blocked his initial appointment, so Kennedy gave him a recess appointment. President Kennedy sent another appointment, and Almond was confirmed 164 days later when Senator Byrd eventually missed a floor session. He took senior status in 1973. By operation of the Federal Courts Improvement Act of 1982 the CCPA was eliminated and Judge Almond was reassigned to the United States Court of Appeals for the Federal Circuit as a senior judge until his death in 1986.