The thing is Kha'zix is an Assassin: He's all about high single-target damage and mobility, which is the meta currently, but he's not very survivable (more survivable than a few others I've played but still.) But yea, if tons of damage was the only thing in this meta, everyone would be playing assassin-type characters. It's not though: There are many elements in the meta, the most potent right now though are speed and power, which Kha'zix and his foe (prey) Rengar are.
If you aren't playing a character with a blink, dash, or speed boost, then you have to be able to hold your ground. Kha'zix may be able to get good damage, but an Assassin's greatest weakness is a Tank, because they survive the initial burst rotation that makes Assassins such nightmares, and after that the assassin can only hope they don't get crushed before you finish with them.
I'd recommend fighter-type or tank-type champions to make an effective counter, like
or use someone with a butt load of CC (which is also their weakness: If they can't get in that initial combo, their power is DRASTICALLY reduced)