Full walkthrough, in case you get stuck (SPOILERS, OBVIOUSLY):
Dungeon -- The princess should glitch out of her cell after the intro, but this may not happen correctly if your framerate is extremely low. If you stay stuck in the cell, select Pause > Retry from checkpoint.
Final boss -- The Gromdak fight is unwinnable because you cannot attack. Let him kill you or jump into the lava. You will respawn in the sky with an error message about an invalid checkpoint. Don't touch any directional inputs and land on the exit platform, skipping the boss fight.
Room 1 -- Just drop from the entry platform and go to the next room.
Room 2 -- When the guard hits you, it will launch you into the air in the direction the guard is facing. Make the guard follow you towards the railing, and then allow the guard's attack to send you over the railing.
Room 3 -- Walk to the left, fall down to the next platform, then drop onto the long logs. Walk across the logs and then drop down to the exit platform.
Room 4 -- Drop off the ledge via the opening in the railing on the right. Then drop off via the right opening again. Drop off the next two ledges. Then drop off from the middle or left opening. Or just run at the ledges and try to angle your fall to skip over the railings. If you get stuck, select Pause > Retry from checkpoint.
Room 5 -- Fall down the elevator shaft and open the chest to get the hookshot. If you step on the switch, the elevator will freak out and may trap you. If this happens, either select Pause > Retry from checkpoint, or, if you want to be fancy about it, hookshot onto the open chest, then hookshot the hallway's wooden ceiling.
Hallway -- The wooden ceiling is a hookshot target. You can hookshot your way over the statues, but guards will attack you. To avoid this, hookshot the higher layer of the wooden ceiling to move to the first layer of the ceiling and then walk to the other side.
Entering the guardroom -- Two possibilities: First, you can just run through the guard room to the room on the opposite side. The guards will only chase you when they can see you, so in the next room, immediately turn the corner to the left to make them lose interest. Alternately, hookshot your way up to the highest layer of the hallway's wooden ceiling and jump off onto the guardroom roof. Go to the opening on the left and hookshot to the ledge across the room, then drop down to the treasure room, avoiding the guards completely.
Big lava room -- The wooden statues are hookshot targets. Throw three wooden statues onto the first platform, hookshot one of them, then repeat the process for the next platforms. Once across, carry the last statue into the cave and throw it onto the switch. The lava will drain. Drop down to the newly opened floor, open the chest to get the jump ability, then chain wall jumps to climb up the vertical shaft on the far side of the room.
Courtyard -- Jump up beside the giant Gromdak statue then fall into the courtyard. Avoiding the guards, jump up to the alcove and then chain wall jumps to get up to the platform at the top. Either jump down outside of the courtyard, or, just for fun, move the statue off of the switch.
Valley -- Open the chest to get the sword, then kill the two guards. Approach the hero to finish the game.