Dapcutscene i pretty much spot on here. I personally like to evolve Q first, as it gives you killing potential if you catch someone out, then evolve E second (ALWAYS evolve this first or second, it's his most crucial evolution), but either way is legitimate, it depends on what the specific game demands.
Evolving R is good, but usually feels like a luxury. Quite often i find myself having to evolve W for the waveclear and the slow, if my team can't bring those things. If you find yourself having to fight groups of enemies, or if you're dying too much, R is a strong evolution to take as the 3rd proc will help you survive quite a bit.
Khaz is very item dependent, and most junglers will beat him early if you fight. Bruisers/AD casters (J4, Lee Sin, Pantheon, etc.) will crush you in the early levels 1 v 1, and until you get some armor pen the more damaging tanks (Sejuani) can kill you too. Khaz does well against other assassins (Shaco [if you ward properly], Yi [IMO]) and tank junglers with lower amounts of early CC (amumu, etc.), but bear in mind that you shouldn't really be looking to fight the jungler 1 v 1 early game. What you want to do is farm like a madman, and gank opportunistically (don't force anything in a pushing lane, DO countergank). Once you have your jungle item, ravenous hydra, and brutalizer/hexdrinker (with a giants belt thrown in for survivability) you will be more than able to annihilate squishies. I personally feel that Black Cleaver is absurdly strong on him, but so expensive that if i can afford it the game is usually won anyway. Just for sustain/burst purposes, hydra is the better first-damage-item pickup.
I hope this helps! Khaz needs a couple of buffs in the League of Tanks, but he's a lot of fun to play :)