Dempster was born in Fremantle, Western Australia, to James McLean Dempster, a sea captain, and Ann Ellen Pratt. He was educated at Lowe's School in Fremantle and at Reverend Dacres Williams' School in Guildford. In the 1860s, he, his brother Andrew and fellow pastoralists Charles Harper and Bernard Clarkson undertook various explorations. In 1864, he and his brother became the first European explorers to reach the Esperance district, and they opened up a stock route to the markets at Perth. By 1866, he had become the manager of the family property, Buckland, at Northam, and on 5 March 1867, he married Mary Bussell, with whom he was to have three sons and four daughters.
