Hammond was born in Washington, D.C., the son of Col. Thomas West Hammond, Jr. by his wife Eileen Hammond, née Bennett. While Hammond's father was a US citizen and an officer in the US Army, his mother was an Englishwoman who had played a role in a theatre play, namely Much Too Shy, in 1942. Hammond has one elder brother, David (b. 1946). Hammond's parents had met and married in London during the War, when his father had been posted there. After the war, the couple had moved to the US permanently, and since the Colonel had a transferrable army job, the family moved numerous times to various army stations across the country during Hammond's childhood. The Colonel, whom Hammond adored, died in 1970.
Hammond was less than 10 years old when he made his acting debut in the Broadway play The Complaisant Lover in 1961. At the same time, he began to shoot for the 1963 film Lord of the Flies, which marked his film debut. After this, Hammond made what was to be his most visible screen role for many years, as Friedrich von Trapp (the elder of the two boys) in the 1965 hit The Sound of Music.
Hammond's next acting role came in 1970, when he appeared in his second Broadway play, Conduct Unbecoming. This was his first role as an adult. In 1972, Hammond appeared as Peter Linder in Skyjacked. In 1973, he made a guest appearance on The Brady Bunch, in the fourth-season episode #090 "The Subject Was Noses", as the high school hunk Doug Simpson who loses interest in Marcia after her tragic football accident. That year he also appeared in an episode of The Waltons, called "The Townie" as Theodore Claypool, Jr. After making the transition from juvenile to young leading man, he spent several seasons in daytime soaps such as General Hospital. He also appeared on many television shows of the 1970s including Hawaii Five-O.
In the late 1970s, Hammond re-joined fellow The Sound of Music alumna Heather Menzies (who played Louisa von Trapp) for one episode of the TV adaptation of Logan's Run. He contributed to the book The Sound of Music Family Scrapbook.
From 1977 to 1979, Hammond played the role for which he is perhaps best known, as Peter Parker/Spiderman in the television series The Amazing Spider-Man. Hammond described his approach to the character: "I liked the idea of taking a fantasy hero and making him believable as a person. I made it clear going into it that I wasn't interested in doing something that was just a camp joke." The series aired sporadically on CBS, with 13 episodes airing over two seasons. A pilot movie appeared in the fall of 1977, with the series returning as a mid-season replacement for five episodes in the spring of 1978. While the show did well in the ratings, CBS was unwilling to commit to a regular timeslot due to its relatively weak showing in the lucrative adult demographic. The second season aired six episodes, each an hour long, in the fall of 1978 and winter of 1979, with a final two-hour episode in the summer of that year. Although Hammond played Peter Parker in the television series, in all of the scenes in which Spider-Man is seen performing stunts or without dialogue, a stunt double was filmed by a second camera unit.
After the Spider-Man series ended, Hammond guest-starred on a number of top-rated TV shows of the early 1980s, including The Love Boat, Magnum, P.I., Murder, She Wrote and recurring roles on Falcon Crest and Dallas.
After being cast as yachtsman Dennis Conner in the 1986 Australian TV miniseries The Challenge about the 1983 America’s Cup challenge, Hammond liked the country so much that he decided to stay. He later became an Australian citizen and has since then appeared in several television miniseries filmed in Australia. This included an important role as an American WW-II officer based in Far North Queensland in the major mini-series Fields of Fire, series I and II, set in the cane-fields of tropical Australia. His character represented the gentler side of the culture-clash between Australians and Americans. He had a starring role, as "Sir Ivor Creevy-Thorne", in Mirror, Mirror, an Australia/New Zealand extended miniseries (a complete story of 20 serialised episodes, with cliffhangers between each of the episodes). Hammond also guest-starred in various Australian television series, including satirical television programs such as BackBerner and CNNNN, and the science fiction program Farscape, and also dramatic series such as The Flying Doctors, MDA and the Australian/USA co-production Mission: Impossible (which was filmed in Australia).
In 2005, Hammond portrayed television producer Aaron Spelling in Dynasty: The Making of a Guilty Pleasure, a fictionalized television movie based on the creation and behind the scenes production of Dynasty.
Hammond is a writer for Australian television, having written both the mini-series A Difficult Woman and the TV movie Secret Men's Business. In 2009 he made his directing debut with Lying Cheating Bastard, a play he co-wrote with magician James Galea.
Hammond married Laura Soli in 1980 and the couple were divorced in 1984. He moved to Australia in the mid-1980s, and he lives in Sydney with his partner, Australian actress Robyn Nevin.
He has remained close friends with his Sound of Music siblings; during their reunion on the 40th anniversary DVD, he said, "You're my best friends in the world".