Complete the game with either character, after you have gotten the two crystals from the Towers.
As Reinhardt, go to the third level in the Tower of Execution, and break open an iron maiden to find an execution key. Return back down to the second level and open the gate with the key. Walk through the hallway and look over the edge to find a floating platform with another iron maiden on it. Jump down to it and break the iron maiden to get the purple jewel for his Second Secret.
As Carrie, head to the top of the Tower of Sorcery and find a floating platform with a torch on it. Put holy water on it then hit the torch to find the purple jewel for her Second Secret.
After that, save the game and then, holding the Analog Stick Up, highlight the character that was used to beat the game. Select the character to get their alternative costumes. Reinhardt's costume is the same as Simon Belmont's costume, and Carrie's costume is a school girl outfit.