NetherRealm has confirmed via a Twitch watchtower stream that the Red Hood DLC for Injustice 2 will arrive June 13th for season pass holders.
Word on the street suggests that those without a season pass will need to wait a week before they can purchase the character individually. The official description for Red Hood reads:
Jason Todd once served as Batman’s second protégé “Robin” before being brutally murdered by The Joker. Years later, the powerful healing waters of the Lazarus Pit brought Jason Todd back to life. However the power of the pit comes at a cost, and a change takes hold in Jason. He now uses mantle of the Red Hood to wage a lethal war on crime.