To get to the inverted castle, you need to do a buttload of things.
First, you need to have the bat creature and have unlocked his supersonic ability, which makes him shoot out a yellow wave like ability to let you see in the dark.
Next, go to the underground, and in one of the right rooms is a room filled with spikes on the floor and walls. Carefully manuever the bat through the room using the supersonic ability (if you hit a spike, SPAM the button to turn back into a bat to allow yourself to not get hurt so much)
When you reach the other end, go through the rooms till you find the spike armor, which allows you to walk on spikes. Equip it and go back (Just walk over the spikes going back and you will destroy them.
Head to the bell towers area, and there will be a room on the far left in one of the bell towers that has spikes on the floor and walls (its really narrow), use the armor to trek through the spikes, then turn into a cloud to pass through the grate blocking the way and continue into the next room. You will get a short cutscene in which Maria asks Alucard to use some glasses on Richter, because she thinks an evil spirit is controlling him. Equip the glasses.
Head to Dracula's room (the place that the game started off in) and fight richter, you will see an orb floating around him, FOCUS ON THE ORB, it takes about 6 - 7 hits to destroy the orb, after the orb is destroyed, watch the cutscene and alucard will automatically head into the inverted castle. Achievement unlocked.