The runway was built in two phases:
The contract for the first phase was awarded in August 1955, and the finished runway was officially opened on 12 Sep 1958, when a Westland Widgeon helicopter's nose severed a ribbon strung across the runway.
In February 1970 the Government approved the project to extend the runway. Construction took longer than originally planned, and the new extension came in to use on 1 Jun 1974.
Source: Chapters 17, 20, and 21, Airport of the Nine Dragons Kai Tak, Kowloon, by Capt. Charles Chic Eather.
The runway numbering '13 / 31' follows the international standard, and is derived from the alignment of the runway. The runway is at 135° or 315° from North, depending on which way you're facing. Runway numbers drop the last digit of the alignment, giving 13 / 31.