Alex was educated at Bedford Modern School, where his father taught PE. He had ambitions to become a cricketer, having captained his school team at county level, before his interest in art at the age of 16. He studied painting at Chelsea College of Arts, graduating with a Bachelor of Arts, and became a member of the Royal Society of Sculptors.
Shortly after college, he was granted a Gilbert Bayes award by the Royal Society of Sculptors to help in his transition to professional practice, following which he collaborated with Conrad Shawcross on his work. After initially focusing on small sculptures, influenced by House designed by Rachel Whiteread and the work of Richard Wilson, Chinneck started working on large scale designs.
Most of Chinneck's earliest public artworks were realised across Greater London. His early works include Telling the Truth Through False Teeth (2012), where the artist used 1,248 pieces of glass to create 312 identically smashed windows across the derelict facade of a factory in Hackney, From the Knees of my Nose to the Belly of my Toes (2013) in Margate where Chinneck created the illusion that the entire facade of house had slid into the garden, and Under the Weather but Over the Moon (2013), a commercial property situated on Blackfriars Road created to look as if it had become completely inverted. For his work in Hackney, local residents have described Chinneck as the "Banksy of Glass".