You unlock The Gamekid by visiting Arcade rooms 10 times over multiple playthroughs.
Arcades can only appear on the second floor of any given chapter (Basement/Cellar II, Caves/Catacombs II etc.). To cause one to appear you must have at least 5 cents on your character going into the second floor.
To enter an Arcade simply walk up to the door and pay 1 cent to enter. Upon your tenth visit the item will unlock.