Drinkwater came from an old Liverpool family and was born on 28 March 1812. His grandfather was James Drinkwater, Mayor of Liverpool in 1810 who married a Miss Elinor Leece, of Ballamona, Braddan. The long European peace, which followed the Battle of Waterloo, resulted in many English children being sent to France for education, among them William Leece Drinkwater, who attended Angoulëme College and became fluent in French. He then attended the Royal Institution School in Liverpool and then St John's College, Cambridge. He graduated with a BA degree in 1834 and later took an MA degree. He was called to the English Bar (Inner Temple) in 1837 and subsequently practised in the Northern circuit acting as a reporting barrister in the Common Pleas Division.