What are the triggers for Goku to become a Super Saiyan in the Parallel Quest 'Legendary Super Saiyan'

Provided that all other conditions are met (i.e. you defeated Krillin before Goku), there is a random chance (I don't know the exact figure but experience suggests it's somewhere around 25%) that Goku will transform. Even if you do get lucky and he transforms, Stage 1 Frieza is an absolute Super Kamehameha magnet and will get thoroughly owned if you don't keep SSJ Goku busy.

Add to that the random factor of the item drops, and you could be grinding for a fair while for the Super Saiyan skill (it might be faster to finish the Android Saga so you can just buy it from the skill shop).

I tried a killion times then I started defeating them in order, piccollo, vegeta, then gohan and of course krillin before goku and he transformed 3X in a row.