Bingham was the son of the headmaster of Dungannon Royal School. In 1926 he graduated in medicine from Trinity College, Dublin. Shortly after he joined the Royal Navy, he volunteered to become a member of the British Arctic Air Route Expedition (BAARE) led by Gino Watkins. He took part as the expedition doctor and to be in charge of the expedition's sled dogs. Together with surveyor Alfred Stephenson and geologist Lawrence Wager he went on a journey to map the ice cap border of the Schweizerland mountains and to attempt the ascent of Mont Forel, the highest point of the range. Back from Greenland he was appointed to carry out an hydrographic survey of the coastal area of the Labrador Peninsula on HMS Challenger. He wintered in Labrador and increased his experience in sled dog teams with the local huskies there.