Here are all the different ways to unlock the different leash colors:
Emerald - Demo exclusive
Crimson - Demo exclusive
Royal - EA Store exclusive
Gold - Limited Edition only
Quicksilver - Online Pass Exclusive
Pulp - DLC
Obsidian - DLC
Flamingo - DLC
In order to change your leash color, you select Barracks on the menu, then Customization, then Custom Leash.