Chapman was educated at Melton Mowbray Grammar School. He showed a strong affinity for science, sports and amateur dramatics and was singled out for attention when a local paper reviewed his performance of Mark Antony in Shakespeare's "Julius Caesar". Graham and his brother John were both avid fans of radio comedy, being especially fond of "The Goon Show" and Robert Moreton's skill of telling jokes the wrong way round and reversing punchlines. Biographer Jim Yoakum said "the radio shows didn't necessarily make him laugh".