Profiles unlock when you put enough points into certain skill trees. By the time you have access to the Tempest and are on your first planet, you should have the necessary skill points available to apply them into your skill trees and unlock Profiles. You will most likely need to be Level 4-5 to have enough skill points to do this. After your third profile is successfully unlocked, you will need to select the profile to 'equip' it. Doing so for 3 different profiles will unlock this.