c. 1300, "carry into effect, fulfill, discharge," via Anglo-French performer, altered (by influence of Old French forme "form") from Old French parfornir "to do, carry out, finish, accomplish," from par- "completely" (see per-) + fornir "to provide" (see furnish). Church Latin had a compound performo "to form thoroughly, to form."
Theatrical/musical sense is from c. 1600. The verb was used with wider senses in Middle English than now, including "to make, construct; produce, bring about;" also "come true" (of dreams), and to performen muche time was "to live long." Related: Performed; performing.