Born in Slough, Clifford came through the Chelsea academy, having signed with them at the age of 8. He turned professional in 2010. He played in Chelsea's pre-season friendly away to Ajax, and on 18 October, he joined the first team for an away Champions League game against Spartak Moscow, where he was an unused substitute. Clifford signed a new four-year contract for Chelsea in September 2011. On 19 March 2012, Clifford made headlines when he, along with Jacob Mellis, set off a smoke grenade and caused an evacuation of the club's Cobham training ground. The pair escaped with a fine after admitting he brought the grenade – understood to be from a paintball expedition – into Chelsea's Cobham training headquarters. The pair were suspended after the incident on 2 March, though Mellis was sacked. On 11 January 2013, he joined Colchester United on a one-month loan deal, and made his debut the next day as they fell to a 2–1 home defeat to Scunthorpe – a club-record ninth successive defeat. Having been joined in Essex by his Chelsea teammate Sam Walker, also on loan, Billy started the remaining four games of his initial loan spell as the U's fortunes began to turn. On 12 February after a 1–0 win at home to Preston North End, the loan deal was extended until the end of the season with Colchester boss Joe Dunne describing Billy as "tenacious, has great technical ability, his heart is big and it's a chance to get out and play which he has taken." A fortnight later he scored his first goal for the club in a 2–0 win at home to Yeovil Town, before assisting Jabo Ibehre to open the scoring in another 2–0 win at Stevenage which took the U's out of the bottom four. He made his final appearance for the club on the final day of the 2012–13 season, helping the club secure a 2–0 away victory at Carlisle United to guarantee League One safety. On 10 July 2013, Clifford joined Football League Championship side Yeovil Town on a six-month loan deal. He made his debut for Yeovil as in their Football League Cup first round victory over Southend United, on 6 August 2013. On 25 August, Clifford returned to Chelsea after failing to break into the Yeovil first team. On 30 January 2014, Clifford was loaned for the rest of the season to Belgian club Royal Antwerp, then managed by former Chelsea striker Jimmy Floyd Hasselbaink.
Later career
On 7 August 2014, Clifford agreed to cancel his contract with Chelsea by mutual consent to seek regular football. On 8 August 2014, Walsall signed the midfielder on a six-month contract. The next day, Clifford made his debut for the club, where he came on as substitute for James Baxendale in the 70th minute, in a 1–1 draw against Port Vale. Clifford made thirteen league appearances for Walsall before leaving the club on 9 January 2015 at the end of his contract. On 15 July 2015, Clifford joined Boreham Wood after a successful trial. He left the Wood at the end of the season after contributing to the club's successful battle against relegation. On 20 July 2016, Clifford joined Crawley Town on a one-year deal following a successful trial. On 6 August 2016, Clifford made his Crawley debut in a 1–0 victory against Wycombe Wanderers, replacing Jason Banton in the 82nd minute. A week later, manager Dermot Drummy gave Clifford his first Crawley start in a 1–1 draw against Doncaster Rovers, in which he featured up until the 70th minute before being replaced by Bobson Bawling. On 5 November 2016, in an FA Cup tie against Bristol Rovers, Clifford scored Crawley's equaliser in a 1–1 draw, scoring in the 35th minute after Lee Brown had given the visitors the lead. On 14 May 2018, it was announced that Clifford would leave Crawley at the end of his current deal in June. On 3 August 2018, Clifford agreed to return to Boreham Wood, signing on a non-contract basis following a successful trial period. After featuring as an unused substitute on two occasions whilst back at Boreham Wood, Clifford opted to join National League South side Hemel Hempstead Town and went on to make his debut in a 2–0 loss against Woking in August 2018. Clifford then followed manager Dean Brennan to Billericay Town in December before joining Slough Town on loan until the end of the season in February 2019. On 20 July 2019, Clifford signed for Wealdstone. He signed a one-year contract extension in July 2020.
