Clarke-Salter first joined Chelsea in 2006 and spent the rest of his youth career there bar one season at Sutton United. He made his under-18 breakthrough at Chelsea in December 2013 whilst still sixteen years old. He became a key figure and the under-18 captain in Chelsea's youth surge after winning the FA Youth Cup twice and the UEFA Youth League. Clarke-Salter was included in Chelsea's pre-season tour of the US in 2015. However, he failed to make a single appearance in Chelsea's disappointing campaign. On 5 March 2016, Clarke-Salter appeared on Chelsea's substitutes' bench in a 1–1 draw with Stoke City. On 2 April, he made his professional debut in Chelsea's 4–0 victory at Aston Villa, replacing Pedro in the 74th minute. After the match John Terry praised him for his performance and attitude, and he also stated that Clarke-Salter has the potential to replace him one day.
2016–17 season: Loan to Bristol Rovers
On 31 August 2016, Clarke-Salter joined League One side Bristol Rovers on a season-long loan along with teammate Charlie Colkett. Despite joining Bristol Rovers on loan, Clarke-Salter stayed at Chelsea and continued to played for Chelsea u23 to regain fitness. Clarke-Salter made his debut in a 1–0 defeat on 28 September 2016 away to Sheffield United. On 18 October 2016, Clarke-Salter played his second match for the Rovers in a 3–3 against Milton Keynes Dons, where he provided an assist for the second goal to the hat-trick scorer Matt Taylor. On 22 October, Clarke-Salter scored his first goal for the Rovers in a 2–0 win over Oldham Athletic. On 22 November 2016, Clarke-Salter was taken to hospital during the 5–1 loss against Charlton Athletic. After the match, Rovers manager Darrell Clarke, stated that Clarke-Salter a fractured his arm and it will be a while before he'll be back. Clarke-Salter made his return from the injury in a 2–0 victory over Southend United. He was an 89th minute substitute, replacing man of the match and goalscorer of the second goal, Rory Gaffney. He made his first start since his return from injury in the 1–0 victory over AFC Wimbledon.
2017–18 season: Return to Chelsea and loan to Sunderland
Following a loan spell at Bristol Rovers, Clarke-Salter returned to Chelsea at the end of the 2016–17 campaign ahead of pre-season and signed a new four-year deal in July. Ahead of the 2017–18 campaign, Clarke-Salter was given the number 35 jersey and made his first appearance of the season during Chelsea's 5–1 EFL Cup victory over Nottingham Forest. On 8 January 2018, Clarke-Salter joined Championship side, Sunderland on loan for the remainder of the campaign. Just under a week later, he made his debut during Sunderland's 4–0 away defeat against Cardiff City, playing the full 90 minutes. On 24 February 2018, Clarke-Salter was shown a red card for a challenge on Middlesbrough's Adama Traore in a match that ended in a 3–3 draw. In his first game back from suspension, Clarke-Salter was dismissed again after receiving a second yellow card in a 2–0 defeat against Preston North End.
2018–19 season: Loan to Vitesse
On 2 July 2018, Clarke-Salter agreed to join Dutch side Vitesse on a season-long loan.
2019–20 season: Loan to Birmingham City
Clarke-Salter signed for Championship club Birmingham City on 24 July 2019 on loan for the season. He made his debut on 6 August in the EFL Cup first round visit to Portsmouth. Manager Pep Clotet fielded an inexperienced team, and Clarke-Salter played the whole of the 3–0 defeat. He did not play in the Championship until 1 October, taking the place of the injured Marc Roberts in the starting eleven for the 1–0 defeat away to Wigan Athletic. According to the manager, "he played with a lot of maturity, he was very focused, he dealt very well with the physicality of the game and worked very well with Harlee as well. He also covered the line very well. I think on the ball he was very good as well so it was a very positive game for him." His performance was similarly positive in his next appearance a month later, when he replaced the suspended Dean for a 1–0 defeat at home to Fulham. He scored his first goal for Birmingham in a 1–1 draw against Millwall on 30 November 2019.
