Are Kakaroto (Goku) and Kaiō-sama (King Kai) the only people that know how to use the Kaio-Ken?
Did Kaiō-sama teach his technique for others beside Goku? For example, while training Krillin and the others.
The exact technique in question is something that only Goku was able to learn and master. King Kai created the technique but never actually had the strength to survive such an ordeal. Goku is his first pupil to actually master it, according to the king himself.
However, to go a little deeper, it's clear that the Kaio-ken is something that King Kai developed in his endless hours observing the universe. Watching not only powerful warriors but never-ending channels of energy like atmospheric pressure, and transfer of water through a planets vegetation (not to be confused with a Vegeta-Tien fusion, lol [wait...eww]) has given the Kai the wisdom and understanding of how to control the flow of ki from intake-consumption-output and recycling. The Kaio-ken technique is a way to control the ki in one's body with the power of a warrior, the control of a Tree, and the sturdiness of a stone, and EXPLODE that energy for just a moment without using the ki which allows the heart to pump and the body to survive.
The spirit bomb is important to think about, too. Unlike most ki attacks, the spirit bomb does not rely on the user's inner ki, but that which he can extract from around him. Which reminds me strongly of Goku's SSJ3 transformation, as well as all power ups, even human ascension. However, what if one were to take in that ki and allow it to flow in the ki-pool inside the body (aka chakra) instead of releasing a huge orb of energy? I think this is how Goku gets his strength, by being pure, all energy is willing to lend its power to him, and once he suppresses his energy and no longer needs the boost, he returns his own energy to the places he borrowed it from.
My point here is that while the Kaio-ken is exclusively Goku's ability, I have no doubt that it contains traces of techniques and powers all around the universe, like the original Super Saiyan of legends, King Yemma in his trip to King Kai's planet, the Namekian race who can use ki for so many things beyond fighting, and so many more life forms that we cannot even imagine.