The gaining of the "Hunt is on" buff is apparently somewhat random. There are two requirements for the buff to come up. 1) Rengar must have the Bonetooth Necklace item, and 2) Kha'zix must have used 3 evolution points (requires him be at least level 16). The exact timing that the buff can start is currently unknown, if it even exists.
If Rengar kills or gets an assist on Kha'zix: Rengar's Bonetooth Necklace will be replaced with the Head of Kha'zix, and will gain a short buff called "Victory!" which as some extra flavor text. The Head of Kha'zix has the same stats as a fully stacked Bonetooth Necklace (regardless of how many stacks it had previously it will be fully stacked as the Head of Kha'zix item) and the stacks won't be lost upon death.
If Kha'zix kills or gets an assist on Rengar: Kha'zix will gain an extra Evolution Point, and gain the "Victory!" buff with some flavor text.