Born in Rainhill, Lancashire, England, A'Court was a winger who started out at Prescot Cables as an amateur before he was signed by Reds manager Don Welsh.
A'Court made his debut in a league match at Ayresome Park on 7 February 1953, a game that saw Liverpool take both the points from a 3–2 win.
A'Court, who followed Rugby league, signed from Prescot as an 18-year-old in September 1952, spurning the advances of Everton and Bolton Wanderers to become an Anfield apprentice, a decision that paid off as just six months after joining he made his first team debut.
During the Anfield club's first season in the second tier of English football A'Court established himself as a first team regular making 33 league and cup appearances.
Although the strong and talented A'Court could have left to play for teams in the First Division, his loyalty to Liverpool was rewarded in 1961–62 when, as an ever-present, he and the Reds celebrated promotion back to the First Division under the guidance of Bill Shankly, finishing a full eight points (in the days of two points for a win) clear of second placed Leyton Orient.
Liverpool had won the first encounter with Icelandic side KR 5–0 but rather than treat the return leg as a mere formality, the Reds finished off the job with a comprehensive 6–1 victory.