Play Castlevania 3 until you reach the Clock Tower and defeat Grant. Allow Grant to join you once you've defeated him. Now, when you begin to make your way back down the tower, make sure you're playing as Trevor when you reach the first set of stairs going down. Walk slowly down the stairs. Now, here comes the tricky part: Press Select to switch to Grant *just* as Trevor is about to walk down off the screen. Trevor's feet should be together, at the very bottom of the screen, and if you did it correctly, once he transforms into Grant, he should automatically walk down onto the next screen. Now turn around and start going back up the stairs, and just keep walking. You'll find that you'll be able to keep going, up past the stairs and off the screen into a glitched-up area of the Clock Tower that you normally could not get into. For some more interesting results, try going back down the stairs one screen, then going back up again. Sometimes, Grant will be repositioned on the "invisible stairs", and thus, be able to go farther with this glitch.
This glitch can also be performed on the first stairway going down in Level 3, the swamp area. This one's even better than the Clock Tower glitchy area, because you can keep this one going longer, and you may even end up seeing graphics from different levels of the game! If you keep going far enough with this glitch, you may eventually end up in a glitched-up version of the boss room where you'd fight that huge Bat, but the boss will not be there, and once you enter the room, there is no way back out of it.
This could work on other staircases in the game, such as the first one on the Pirate Ship and the last one before you get to Alucard's Boss room.
Taken with permission from Flying Omelette.