Souray was born in Elk Point, Alberta, to Richard Souray and Lillian Parenteau. He is of Métis descent and has one sister, two years older. At a young age, Souray moved with his family from Fishing Lake Métis Settlement to a small community in Alberta in order for his sister to attend school. In another two years, they moved again so that Souray could play organized sports. While still in elementary school, Souray's parents divorced; he first lived with his mother before moving in with his father at age nine. Growing up, he attended Archbishop O'Leary Catholic High School with former fellow Oiler Fernando Pisani. Souray has a tattoo of a Métis Chief in full headdress on his arm. He says the tattoo is meant as a reminder to himself to be a positive role model for Native and Métis children, as he himself is proud of his Métis heritage.