To unlock Kunio and Ricky, you need to beat the game once. After that, they will be available at the character select screen.
You don't need to play the whole game with Kunio or Riki. You just need to beat the final boss Sabuko (again). All you need is a second controller. As long you didn't start a "New Game+" with your save game you can select the "loiter" option at the save game screen and start the final boss battle again with co-op active.
Select either Kunio or Riki with your account and use your primary character with the second controller and beat Sabuko. You don't even need to survive that fight as Kunio or Riki.