When the fight begins, roll up the ramps, staying close to the edge to keep from getting slowed down by the walls. If you roll quickly, you will easily out run Helmaroc King. Once you get to the top, select one of the following strategies according to the conditions of your run.
Note: It is also possible to preform a storage-less superswim in the water at the beginning of the room and to build up enough speed and momentum to carry you all the way up the ramps and behind Helmaroc before he has time to land. This is, however, only really viable in a TAS.
This is probably the easiest method of skipping the Helmaroc King. To start, make sure you have 1.5 Hearts or lower for your health (This is important because it slightly changes Link's hitbox for the clip to work). Pick up one of the blue pots near where Helmaroc attempts to block you from going up. Walk in and out of the trigger that changes his position from being in the air to him blocking you. Doing this causes him to be manipulated weirdly and he flies away from the arena. You have to keep walking in and out of the trigger until Helmaroc's shadow disappears from in front of you. After it disappears, go in and out three more times, then press the B Button to drop the pot and roll up the ramp so that you'll be sandwiched between Helmaroc and the wall next to you when he flies back to his position. Now just climb up the platform, and Helmaroc will automatically clip you into the floor. From here, just crawl under the arena's spikes and jump to the ring of collision around the arena. From there, you can just roll up the pathway to Ganon's door.
Note: Due to the difference in Helmaroc's hitbox between Wind Waker HD and original version of the game, this trick is much harder in the original version. In The Wind Waker, attempting to sandwich yourself between Helmaroc and the wall will not work. You have to grab the ledge and begin climbing it before the Helmaroc King returns to his position. This can be achieved by running in and out of the trigger more and making Helmaroc go further away.
To perform this method, you first need to leaf around the Helmaroc King, then damage yourself down to 1.5 Hearts of health or less using the spikes that surround the arena. Now, you need to jump in the giant hole in the arena, use the leaf to fly to the ledge that is closest to the spikes that prevent you from leaving the arena early, and grab the ledge. Grabbing the ledge correctly, however, is trickier than it sounds. It is recommended that you allow the wall on your right side to guide you to grab the very right side of the ledge. At the very last moment, turn link slightly left and release the leaf as late as you can to grab the ledge in the correct spot. If done correctly, you can hold into the wall and you should automatically clip into the floor of the arena. As mentioned above you want to damage down to 1.5 hearts or less, which is the threshold for critical health, as this slightly alters Link's colllision. This is what makes the clip happen automatically if you ledgegrab at the right spot. If you are not damaged down, or cannot quickly do so (like in a 100% run), you are still able to roll clip from the ledgegrab. This makes it frame perfect, forcing a retry everytime you fail. Now, put the leaf away and crawl under the wall, then just simply jump to the pathway and make your way to Ganon's door.
Below are alternate setups that avoid inconsistencies due to targeting helmaroc or sidle movment. The latter two save ~3 seconds over the first setup made by sva16162.
After hitting the Helmaroc King with the skull hammer on the ramp, it is possible to roll clip onto the platforms that usually close when you activate the second part of the fight. If you successfully roll clip, you'll be halfway into the floor of the main arena. Now you need to crawl under the spikes that surround the arena, and using the skull hammer, jumpslash onto the ring of collision around the arena. When you stand on the ring of the arena for more than about 1 second, the cutscene for the battle teleports you back into the arena. To avoid this, you need to backflip and pull out the leaf to move forward on the ring of collision. Once you make it to the path past the arena, sidehop and backflip until you reach Ganon's door. You can easily open the door by sidehopping into it and then mashing A to open it.
First, leaf to the ledge behind Helmaroc. After this, take up the rock nearest the spikes on the right side, since this one contains hearts. Make sure you only have one quarter of a heart left. Line up to the spikes and place the rock behind you. Take out a bomb and place it in front of Link, between him and the spikes. Turn around quickly, take up the rock, face the spikes and throw it. If lucky, one or more of the hearts will get on top of the spikes. If so, let the bomb kill you and zombie hover towards it. If you get it, you can jump of the spikes and run up to Ganons door.
Note: The hearts in this trick are totally luck based and might not land on top of the spikes 100% consistently.
Once at the top of the spiraling walkway, sidehop and use the deku leaf to grab the platform behind the Helmaroc King. Align yourself properly, and zombie hover towards Ganondorf's look-out. Hover to the loading zone behind the door and proceed.
Start off by leafing to the ledge, just like the short hover method, to skip hitting Helmaroc. Then head towards the spikes that blocks the path leading to Ganon. You will want to sidle the wall on the left of the spikes to get a consistent position, then drop a bomb and jump on the spikes. Proceed by performing a well-timed sidehop with the right angle, then use the leaf to glide a little bit, and perform a jumpslash right before the bomb explodes. This will boost you up above the spikes, and from there you can head towards Ganon.
Here is the original method:
After you have pounded Helmaroc King with the Skull Hammer and he has fallen, pull out the wind waker and get dry storage off of the ledge in front of you. Now climb up to the top and trigger the cutscene. Due to the fact the you preformed storage, you will be able to walk around during the cutscene. Now jump back down to the bottom where the battle began before the top of the tower closes (which still costs a 1/4 heart).
Once the cutscene has finished, pull out the wind waker and then press the button it is on again to fix the camera. Roll up the stairs and get on top of the first wooden cell by the stairs while waiting for Helmaroc.
Note: This is the risky part of this strategy — there is a high risk that after the cutscene of the roof closing ends, Helmaroc often gets stuck in the sides of the tower while trying to get to you. Due to this, this strategy is probably best utilized in a segmented 100% run.
When Helmaroc lands and comes over to peck you, due to the fact the you are on top of the cell and he cannot reach you he will continue walking forward infinitely. This means that you can even pound his head with the Skull Hammer until his mask falls off and he will continue to walk forward. Be careful after his mask falls off, as the properties of the cutscene will clip you into the wall. From here you can simply use quickspins to finish him off.
After the cutscene of Helmaroc's death is over, you will automatically be warped on top of the tower — free to grab the heart container and proceed.
After the third time you have hit Helmaroc King with the Skull Hammer, lure him into the corner near the spikes. After he pecks at you, and gets his beak stuck in the ground, hit him once more with the skull hammer. This will trigger his 'unmasking' cutscene, which automatically warps you to his head. This clips you past the spikes allowing you to skip his second phase.