His father Yannis was a timber merchant and wanted him to attend university and then go into the family business, but Kokkinos' prowess in soccer had led Lefter Küçükandonyadis to their door.
However, in 1964, the Kokkinos family were forced to move to Athens, in the midst of the Cyprus dispute, where Turkish Prime Minister İsmet İnönü renounced the Greco-Turkish Treaty of Friendship of 1930, and took actions against the Greek minority.
He failed to settle in Athens and at sixteen years old, decided to purchase a ticket to tour the Patris which carried migrants to Australia.
When the Patris stopped at Fremantle, Kokkinos left the ship to an address occupied by fellow Greeks and was given a change of clothes and a train ticket to Melbourne.
Hungry and broke, he sat outside a Greek restaurant on Lonsdale Street for hours on end, where a daughter of a local priest took pity on him.
Kokkinos went to see South Melbourne Hellas, and attracted attention within twenty minutes by performing keepie uppies.
When Kokkinos refused to cut his hair prior to a match against Melbourne Hungaria, he was forced to have a haircut at half-time.
[10] By June 1974, he was back at South for the fourth time in his career,[11] and finally won the State League, as well as that year's Dockerty Cup.
[10] Kokkinos was then signed by West Adelaide Hellas as a replacement for John Kosmina, who had joined English First Division club Arsenal.
Kokkinos was having an affair, and in order to fund a lavish lifestyle and a gambling addiction, he threatened to tell journalists of their relationship if she did not provide him money.
Kokkinos fathered another son, and later settled in Mordialloc, still keeping in touch with former players and attending South Melbourne's home matches.