Henderson signed for Celtic in 2008. He featured for the club’s development squad in midfield and was included in the first team squad for their pre-season trip to Germany in the summer of 2013. Henderson played in a pre-season game against Ukrainian side Sevastopol, alongside higher profile debutants Virgil van Dijk and Amido Balde.
Season 2013–14 saw Henderson continue to play for the Celtic development squad at both under-20 level and in the UEFA Youth League. He was particularly impressive in Celtic’s 4–1 win over the Ajax academy side in October. Henderson made his competitive first team début, aged 17, on 6 December 2013; coming on as a substitute in Celtic’s 5–0 win over Motherwell in the league. He then featured for the first team again in January 2014, playing in Turkey against Galatasaray in the Antalya Winter Cup. The match finished 0–0, but Celtic lost on penalty kicks. Despite the result, Henderson, and several other youth players who featured, were praised for their performance.
On 13 March 2014, Henderson made his first start in a competitive game in an away league match at Kilmarnock. Celtic won 3–0 and Henderson was praised by manager Neil Lennon for his performance, saying “I thought he grew into the game and had a superb debut. Credit to him and credit to the academy, because I think we have got a good player there.” Henderson scored his first goal for Celtic on 26 March 2014, the second goal in Celtic’s 5–1 win away at Partick Thistle which clinched the league title. His form throughout March saw him win the SPFL Young Player of the Month award for that month. Manager Neil Lennon commented “He’s been absolutely excellent”, and added that “His attitude to work is fantastic and there’s certainly a willingness on his part to get better”.
Henderson made a couple of appearances in the first few weeks of season 2014–15; playing against KR Reykjavik in a Champions League qualifying match and then in a league match away against Inverness CT. However, he thereafter completely dropped out of new Celtic manager Ronny Deila’s plans for several months. Henderson regained his place in the XI in early January and scored an excellent left-footed strike against Hamilton Academical on 17 January 2015. Henderson featured for Celtic in the first leg of their Europa League tie against Inter Milan, coming on as a substitute in the second half and providing the cross which led to John Guidetti’s late equalising goal in a 3–3 draw. A month later he played in the final eight minutes of Celtic’s 2–0 League Cup Final win over Dundee United.
On 14 August 2015, in between his two spells away on loan, Henderson signed a three-year contract with Celtic.