Russell was born on April 26, 1812 in Glasgow, Scotland, United Kingdom.
Although christened William Thompson Russell Smith, he was known simply as Russell Smith. His father, William Thompson Smith, and his mother, Margaret (Russell) Smith, a practising physician who had studied medicine in Glasgow, emigrated to America with their children in 1819 because of their political views. They settled in Indiana County, but moved to Pittsburgh in 1824. There Russell's father, who is said to have been "an ingenious mechanic, excelling in the manufacture of cutlery, artists' tools, and mathematical instruments", established a cutlery business.