In order to unlock Green Loontern, you have to complete all of his quests. Around halfway through the game, you'll find him standing in the Watch Tower near a doorway. Complete the quest and you'll have access to the Moon hub, which allows you to transport to the seven Lantern worlds (which become available after beating their respective stages in Story mode).
All of the Lantern worlds (aside from Oa) have introductory quests where Loontern shows you around the planet. Each planet has four additional quests with several of them being Loontern quests. You'll know when you've completed all of Loontern's quests on that particular planet when the next quest giver is not Loontern.
Once you've completed all of his quests, you can purchase Loontern for 50K studs. Keep in mind that there is only one Loontern quest outside of the Lantern worlds and none on Oa so just keep doing moon quests until Loontern isn't showing up there anymore.
Once you've unlocked him, the best place to travel is Oa, which has 10+ races (they're shown as giant rings with a gold brick/stud in the middle). Simply complete one and this achievement will unlock.