In the anime, Killua confesses that he doesn't have a purpose in
life, and what he desires most is to have a friend with whom he
can share adventures.
So, he helps Gon with his purpose (which was to find his father).
In the process, Killua wants to become stronger and compete in a
healthy way with Gon since he had decided not to be part of his
family's business.
Their relationship begins when they are very young, as Killua
discovers that Alluka harbors a dark creature named Nanika, which
emerges into the real world when three of Alluka's requests are
fulfilled. In the end, when Gon requires a lot of help to heal, Killua
decides to ask his sister for a favor so that Nanika can heal him.
After helping Gon reunite with his father, Killua decides to protect
his sister and free her from the control of their family.