Craig was a Sixth Form student between 2004-06 having completed his GCSEs at King Arthurs School, Wincanton. He was a popular student and played a few games for the school first 11, although by then he had already started his career with the Yeovil Town youth system in 2004. At the start of the 2005–06 season he was awarded a first team contract and had three short-term loan spells in the 2006–07 season with Weston-super-Mare, Taunton Town and Tiverton Town. He made his first team debut for Yeovil Town as a substitute away at Gillingham in the final league match of the 2006–07 season.
After making a breakthrough and challenging for the number one right back slot in the 2007–08 season, he became a regular starter during the 2008–09 season. In May 2009, he signed a new 2-year contract with Yeovil Town, and after the current captain was dropped due to injury and poor form, was made captain for the remainder of the season.
Despite being offered a new deal at Yeovil, in July 2011 Craig signed for Peterborough United on a three-year deal where he made 112 appearances in all competitions over the three seasons, including winning the Football League Trophy in 2014.
In July 2014, he signed a two-year deal with Sheffield United and started as a regular in the side both at centre half and then in his more customary role at right back. In his second season at United, Alcock found match time hard to come by, and as a result joined fellow League One side Doncaster Rovers on a 28-day loan in November 2015. The loan was made permanent in the January 2016 transfer window.
Unfortunately he suffered an injury in 2017 pre-season training and has only played 450 minutes this season, but is hopeful for more match time next year.