There are several methods that can be used to head to and from Zora's Fountain without talking to and moving King Zora, allowing you to skip the Ruto Bottle.
By crouching and shielding an explosion, Link gets pushed backwards a bit. By setting this up so that you are at the acute angle in Zora's Domain, you will be pushed through it and end up in Out of Bounds water. From here you can simply swim past King Zora and head to the Fountain.
It's possible to stand on the very corner of the platform in front of King Zora and do a mega-sidehop (similar to a Megaflip) to clip out of bounds and swim past King Zora. You will need to jumpslash immediately after you clip so that you don't just keep going backwards out of bounds.
By setting up a Bomb Hover, you can hover over King Zora in about 5 explosives. Note that this method is slower than the others and requires more explosives, so it is rarely used.
Ledge cancel can be performed with a bombchu or bomb, then you can simply walk through him. You must sidehop or backflip off the small ledge near King Zora because ledge cancel will be cancelled if you jump normally.
Using an acute angle clip, you can skip past King Zora as adult using only the Master Sword. Hold right once you clip to stay out of bounds, then hold up-left to jump into the water and swim up behind him.
If you are returning from the fountain, King Zora will still be in the way. You can do the above method and hover over him again. However, as an adult there is another option if you have blue fire and he is still frozen. It is possible to clip through him as he is melting. There is a short time window when you can just walk right through King Zora and return to the front.
With this trick, you can get the Zora Tunic and complete the trading sequence without obtaining a bottle or entering ice cavern.
At this point, King Zora will be unfrozen as soon as he is reloaded. The fastest way is just to walk down the hallway and back up.
The following setup will not work if King Zora has already been moved by using the Ruto Letter.
See Early Eyeball Frog for more information.
In Zora's Domain, you can clip out of bounds using the ladder next to Zora River entrance. As an Adult, this clip can be used to reach both the Lake Hylia shortcut and the Zora Shop.
Using a well placed sidehop, it is possible to clip without the hookshot. This is shown at the video below at 0:35
Using the Lake Hylia Shortcut greatly cuts out the time spent delivering the frozen Eyeball Frog portion of the trading sequence.
Perform the Ladder Clip, then once out of bounds you'll need to get beneath the ice in order to reach the shortcut.