Larbert, Stirlingshire born inside right Billy Grassam began his football career playing for Scottish Junior Football Association sides Redcliffe Thistle and Maryhill before joining English Second Division club Burslem Port Vale in July 1899, making his Football League debut that September at New Brighton Tower. Playing at outside right, he scored 6 goals in 35 appearances for The Valiants during a single season at the Cobridge Athletic Ground, including scoring on his home debut in a win over Middlesbrough. He then signed for Southern League club West Ham United in the 1900 close season, where he was a regular goalscorer over the following three seasons, netting 44 goals in 83 matches for The Hammers.
Signed by Scottish League club Celtic in May 1903, he made his Celtic debut in their 5-0 Charity Cup Final thrashing of Hibernian on 21st May 1903, but only went on to make 2 further League appearances for Celtic although in his second match Celtic wore hoops for the very first time. By September he had returned south to sign for Second Division club Manchester United, where, playing at inside right or centre forward, he scored a dozen goals in his debut season at The Red Devils finished third in the division, becoming joint-top scorer in the League with Billy Griffiths and Tommy Arkesden. However he had lost his place by the start of the new season and only scored twice in 9 matches during their 1904-05 campaign before a move back to Southern League football with Leyton in July 1905 after 14 goals in 37 appearances for Manchester United.
Grassam soon returned to West Ham United in December 1905 and enjoyed four further seasons at Upton Park, scoring 24 goals in 96 further appearances for The Hammers, Having lost his first-team place to Danny Shea, Grassam moved on to league rivals Brentford in 1910 and played 4 matches for The Bees before the end of 1909-10, after which he retired. Grassam was the second-highest goal scorer behind Shea for The Hammers of the period before the club joined The Football League in 1919.