Catherine

Stage 9 is The Empireo, the final stage. It has 6 levels, the latter two of which are boss levels. It introduces mystery blocks and monster blocks. If you prefer video walkthroughs, go here:
9-1: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...
9-2: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...
9-3: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...
9-4: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...
9-5: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...
9-6: http://catherinethegame.wikia.com/wiki/Walkthrough:The_Empir...

If you'd rather find the path through each level yourself, I've provided some tips here.

About the enemies:
This stage includes monstrous sheep with huge axes on the blocks with you. Unlike sheep in previous stages, they can kill you if you are adjacent to them, and they can jump up two blocks. You can still push them down; do so if one gets close.

About Monster blocks:
Monster blocks can move by themselves, and they will bite to make you fall if you are hanging from them. You can disable monster blocks by jumping onto them. They're tricky because of their unpredictability; they can move out of the way as you jump onto them, block your ascent up a section or take away your tower's support.

About Mystery blocks:
When you step on a mystery block, it will turn into another kind of special block at random (but not a Vortex block). It's often just a heavy block or cracked block, but it can be much worse.
Mystery blocks can become bomb blocks, which pose a danger of destroying your staircase. Don't step on a mystery block until you have to.
Mystery blocks can become ice or trap blocks. If you step on a mystery block, immediately hang from its edge just in case.

About the 9-5 boss:
The penultimate boss has three attacks.
His kick is the easiest to avoid; when you see him lining up for it, don't be hanging from a block facing him.
His gun is more problematic; it can destroy blocks you were intending to climb. Watch the dot that indicates his aim. The dot sweeps left, right, and left again, and when it reaches its leftmost point the boss locks onto Vincent's current location to fire. This is plenty of time to draw his aim to a specific spot and then move out of the way. He will usually fire three times, but sometimes only twice.
His gavel is the trickiest attack; it changes all special blocks on the stage to another kind of special block. The first time he uses it, it changes them to bomb blocks; the next time, dark blocks; then ice blocks; then cracked blocks. This doesn't affect blocks that have not yet fallen onto the stage.

About the 9-6 boss:
The final boss also has three attacks.
His laser beam outright destroys the blocks in its horizontal path. He only aims it lower down the level though, so if you're moving at a decent rate you won't be in danger.
His curse is the worst one; while this is affecting Vincent, pulling any block will cause all special blocks to shift type according to the following pattern:
Dark block; bomb block; cracked block; heavy block; trap block; ice block; monster block; spring block.
If you can maintain your awareness of this pattern in addition to the stairs you build, you can even use the spring blocks as shortcuts.
The third attack, the meteor shower, doesn't occur until you reach approximately the two-thirds mark of the level. Rocks will fall onto the level, turning the block they hit into a cracked block or outright killing Vincent if he's hit. Watch for the red glow on affected blocks to get out of the way.