To unlock Guppy's Hairball you must transform your character into Guppy during one of your playthroughs. To achieve this you must collect three separate Guppy items over the course of your run. Items that count towards this are
Guppy's Tail
Guppy's Head
Guppy's Paw
Dead Cat
Guppy items are primarily gained from either Deals with the Devil rooms or from Red Chests found through the floors at random or in Curse Rooms.
Deals with the Devil are triggered randomly on each floor but there are ways to further your chances of one turning up, the best and simplest way is to make sure you do not take and Red Heart damage during a floor or during the boss fight, to help ensure this happens Spirit Hearts help greatly as losing HP via Spirit Hearts does not affect your Devil Room chances.
You do not have to be holding all three items at once for the transformation to take effect, so long as you have picked up three of the necessary items over the course of your run you will become Guppy.