605 Squadron were not equipped with spitfires during WWII. Hurricane, Havoc, Boston and Mosquito.
First identify exactly what type of aircraft it is. Then go to the Key Publications Forum and ask your question there.
http://forum.keypublishing.co.uk/forumdisplay.php?f=4It would help considerably if the serial number of the aircraft can be seen in the painting. If not then unless the aircraft is well known you will have many difficulties.
If you know the serial number then you can ask the RAF Museum Hendon for the Form 78 Aircraft Movement Card. That will tell you when and where it was assigned until it was Struk Off Charge (SOC). After that you can go to the National Archive at Kew and ask to see the Squadron Operational Record Book (ORB). With a bit of luck, in there, you may be able to find the serial number and pilot(s) that flew it listed. There maybe someone on the Key Publishing Forum that as done this already or has copies of the documents and save you the trouble.