5 - 1965 model year
2 - Hamtramck assembly plant
113 30165 - Shipping Order Number (SON) - This number should be on the body tag attached to the inner fender on the driver's side.
The "113" is Scheduled Build Date - January 13, 1965
You might check with Chrysler Historical to see if they can look up the build record with the SON. If they can do it, the build record would have both the VIN and the SON. That might help you out with the DMV. The number you found is a unique number and no other car built will have it.