To become a Super Saiyan, you first need to be a Saiyan. Then level up your character to level 40. Next, you must fight Vegeta in the Capsule Corp. time rift. Fight and defeat him a second time in that area and you'll be able to obtain the Super Saiyan ability from the Saiyan Awakening mission. You must speak to Bulma and Trunks and defeat both Vegeta and Goku at the same time, while they are in their own Super Saiyan forms.