The Story of Rengar
Rengar was born a bastard among vastaya — neither fully accepted by the pride-societies of his kind nor by the human communities nearby. He resolved this early by deciding that status among either group was less important than being undeniably the greatest hunter alive. He has pursued that goal with the single-minded intensity of someone who has nothing to fall back on if the goal fails — because he genuinely has nothing to fall back on.
His hunts have taken him from the Kumungu jungle across Runeterra's most dangerous territories, always seeking prey worthy of his skill. He catalogs his kills on his body, in scars and trophies that mark each significant achievement. The problem with being the greatest hunter is that the pool of genuinely challenging prey keeps shrinking. Everything that doesn't fight back hard enough is a disappointment. Everything that fights back hard enough nearly kills him. He finds the latter acceptable.
His encounter with Kha'Zix — a Void predator that genuinely challenged him and left him with a ruined eye — is the event that currently defines his purpose. Kha'Zix represents something he hasn't been able to kill yet, and the existence of an uncompleted hunt is something Rengar cannot leave alone. He is not hunting Kha'Zix for revenge, exactly. He is hunting it because it's the only prey left that demands his absolute best, and Rengar cannot function without something that demands that.