The best way to repair a tire puncture is tedious, difficult and requires expensive equipment. Some punctures are non-repairable due to their location on the tire.
One should use a proper wheel mounter/dismounter unit with plastic no-marr grabbers. The tire should be inspected for internal damage or air/water intrusion into the belts. The repair patch should be properly vulcanized. Then the whole assembly should be balanced on a road-force machine.
(Run-flat tires have their own separate issues and considerations.)