Buxton was born in Liverpool on 18 August 1934, one of two children born to Harry and Violet Buxton. Her grandparents on both sides were Jews who had immigrated to England from Russia. She was raised partly in South Africa. Buxton's father owned a successful cinema chain in northwestern England, which allowed her to attend boarding school at Gloddaeth Hall. While there, a coach noticed her tennis ability and urged her to acquire more training.
Buxton began playing tennis at a young age. After spending time undergoing training in London and Los Angeles, in 1954 she earned the British No. 4 ranking. Buxton then reached the 1955 Wimbledon singles quarterfinals and climbed to World No. 9 in the rankings. She played in Wightman Cup competition for the United Kingdom in 1954, 1955 and 1956.
Buxton had her most successful tennis year in 1956. She won the women's doubles title and reached the singles final at Wimbledon. She won the English Indoor and London Grass Court singles championships and the English Hard Court doubles crown (with Darlene Hard). At the French Championships, she reached the singles semifinals and won the women's doubles title with Althea Gibson, who was the first African-American champion. Buxton was ranked World No. 5 by World Tennis Magazine and World No. 6 by Lance Tingay of The Daily Telegraph and the Daily Mail.
Buxton also won the women's singles title at the Maccabiah Games for Jewish athletes on two different occasions: 1953 and 1957.
After suffering a serious hand condition in late 1956 (tenosynovitis), Buxton was forced to retire following the 1957 season at the age of 22.
Buxton's Judaism played a role throughout her career. Her religious background prevented full acceptance within the tennis world from an early age. Regarding her experience at the Cumberland Club, she said:
I had to fill in a form: name, address, telephone number and then religion. I had several lessons there with a guy called Bill Blake, and I kept asking him about membership. Eventually he turned round to me and said, "Look, Angela, please don't keep asking me, you're not going to be able to join the club." I said, "Why not? I'm not good enough?" "No, because you're Jewish." And that was the beginning. It was the first time it (prejudice) had hit me in this country.
Buxton was repeatedly refused access to training facilities because of her ethnicity. From the mid-1950s, she was able to practise at the private indoor court of Simon Marks, the Jewish owner of department store chain Marks and Spencer, who had become aware of the difficulties which Buxton faced.
No Jewish tennis players were admitted to the All England Lawn Tennis Club until 1952. Buxton said in 2004: "I think the anti-Semitism is still there. The mere fact that I'm not a member is a full sentence that speaks for itself." The Chairman of the Club said he could not comment until he investigated further. "I wish it still wasn't such an elite sport", Buxton said. "I wish we could bring it down to a common baseline. It's going that way. It's still not there." After Gibson and Buxton won the doubles at Wimbledon, one British newspaper reported the event with a story titled "Minorities Win". "It was in very small type", said Buxton, "lest anyone should see it".
Buxton was inducted into the International Jewish Sports Hall of Fame in 1981. In 2014, she was inducted into the National Jewish Sports Hall of Fame. In 2015, the Black Tennis Hall of Fame inducted Buxton, honoring her for her doubles partnership and friendship with Althea Gibson as well as her efforts to raise funds for the ailing Gibson near the end of her life.
Buxton wrote the tennis books Tackle Tennis This Way, Starting Tennis, and Winning Tennis and Doubles Tactics.
SR = the ratio of the number of Grand Slam singles tournaments won to the number of those tournaments played.