Hello, I'm NicoB967 and I am currently gold on EUW, I know it doesn't sound pro but I spend much time playing Kha'Zix and I definitely know something about him. I'm not a native speaker so please forget my mistakes :)

Passive

Kha'Zix passive is useful for ganks, it provides a strong slow and bonus damage. Try to synergize it with brushes and your R to deal more damage in teamfights.

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Q, Taste Their Fear

Kha'Zix Q makes him a nightmare for the carries, try to jump on the enemy ADCs in teamfight if they're not pretected to deal tons of damage! If an enemy is near minions he's not considered isolated :( Max this first and evolve first as well.

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W, Void Spike

Useful ability to recover some health in jungle or to slow the enemy, max this second!

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E, Leap

It's your most important ability! It doesn't deal much damage but a wise use of your leap makes the difference between taking a kill or dying. Evolve this second due to the reset on kills and assists that makes you unstoppable!

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R, Void Assault

Your R allows you to evolve your abilities (in order: Q,E,W/R) and gives you the possibility to become invisible for a short period of time! Your R is AWESOME!! With a clever use of it you can proc 2/3 Qs and passives to your enemy without being touched!

The normal jungle routes are:

Starting on the blue side:

Smite krugs, get the red buff and go to the blue buff(smite it)! After your route, kill the scuttle crab and look if you can gank top or mid, if not keep jungling to get money!

Starting on the red side:

Smite the gromp, get the blue buff and go to the red buff(smite it)! After your route, kill the scuttle crab and look if you can gank top or mid, if not keep jungling to get money!

With Kha'Zix it's important to kill the little monsters first to get the isolated bonus on the large monsters! Remember to spam your W to deal area damage and heal yourself.

Your early game with Kha'Zix should be focused on farming, but if there are any possibilities to get a succesfull gank, do it! Remember to CHANGE your early routes if you are vs Shaco, he will probably wait for you near the further buff.

As you get your level 6 you become a killing machine, gank where you can get kills and get early objectives, as far as you keep going like this and finish your core items you become a monster. If you are almost full HP and there is no danger you can also get a solo dragon. If you can try to group up with your top laner to kill the rift Herald and get the control of the game. In mid-game skirmishes(2/3 vs 2/3) jump and kill the enemy carry if you can, but NEVER engage first and be sure that they're isolated. If not just support your team and use your E to escape if things get bad.

In late game you are almost full build and you have an insane damage!! Try to lead your team to victory by killing the isolated carries and jump out immediately! Keep control of the dragons and baron, they are easy to steal with kha, you just need to jump over the baron/dragon spot and press Q and smite almost together to deal insane damage to the isolated objective! Get out with flash if your team doesn't follow or try to kill someone to get the E reset.