late 14c., compleinen, "lament, bewail, grieve," also "find fault, express dissatisfaction, criticize," also "make a formal accusation or charge to an authority," from stem of Old French complaindre "to lament" (12c.), from Vulgar Latin *complangere, originally "to beat the breast," from Latin com-, here perhaps an intensive prefix (see com-), + plangere "to strike, beat the breast" (from PIE root *plak- (2) "to strike").
The sense evolution is from "expression of suffering" to "grievance; blame." The transitive sense of "lament" died out 17c. Also from late 14c. as "utter expressions of grief or pain," hence, chiefly poetically "emit a mournful sound" (1690s). Related: Complained; complaining.