You seem to be looking for advice on lane Khazix, which i don't play much. I do play a lot of JUNGLE khazix, so here's my advice which i gave in another top asking for advice (i'll give my thoughts on your questions in another comment):
I play Kha'zix a lot, so here's my advice:
For runes, i take AD quints, AD reds, Armor Yellows, CDR blues. Some people take some MR per level blues and those aren't bad, but honestly Khaz's ability to clear the jungle has been nerfed so often you need the CDR to survive. It may be tempting to go MS quints or Armor Pen reds, but again, in the early game you need the flat damage just to clear effectively. You will have enough armor shred through items later on.
I actually recommend starting Q, but max W. Q, even at level one, has a really short cooldown, and the heal+AoE from W missiles is vital for your clear. If you know you aren't ganking at level 3, take a second point in W rather than your leap, E.
As for evolutions, the way Khaz is right now the first two you evolve should be your Q and your E, every time. Even though i recommend maxing your W, it is usually better to evolve your Q first; if you catch someone alone, it does pretty incredible damage, and it also increases your AA range. If you're having a really hard time getting ganks, or your lanes already have great kill potential, evolving E first is viable. Evolving this gives fantastic range to his gap closer, plus gives the resets. If the E isn't evolved by level 11, something has gone wrong.
Most people suggest evolving W (missiles) third, and this is a good choice; it greatly increases the effectiveness of your missiles and adds a slow, and you should definitely evolve this if your team lacks waveclear or sticking potential. Personally, i prefer evolving his ult (R) if i'm doing well; it really helps his survivability in teamfights, and helps him escape tricky situations more easily. If you aren't doing so well, W will help your team more.
For clear, i usually start gromp/krugs with bot lane help, then take nearest buff, then go immediately to your next buff and take it (your 2nd charge of smite should be up to help you kill it). Recall and buy (ALWAYS pick up at least one ward to place in the enemy jungle or in the river), then continue your clear. Your ganks will be fairly weak, so only consider it if
- The lane opponent is really low and not pushed to their own turret
- Your team's laner has CC to lock/slow them down
- The lane opponent is pushed and really out of position.
I recommend not trying to gank bot lane until level 6; your kit is better against isolated targets, and you are super squishy until you get some items. Also, be careful if they have a tank jungler; you can try some risky plays in the enemy jungle if they have a fiddle or nocturne, but if they have a sejuani or nautilus you will just die when his teammates come to help.
Your ganks will be fairly good at level 6, and by the time you hit level 11 you should be a terror.
Unfortunately, the strongest item to rush for him, the brutalizer, now only builds into Youmuus, which i find to be sub-optimal for him (still builds into warrior enchant, at least). A strong build for him is:
(challenging smite also acceptable, or trailblazers if you fall behind)
(you should get tiamat as soon as possible; the sustain and burst are invaluable)
(or merc treads, if they have crazy CC)
(feels awkward getting a phage, but this item is still really strong on Kha)
(you will need at least one survivability item if you don't want to get blown up late in the game, and this is the best all-around one for him)
If they have AP carries that are getting fed, an early 
is a solid acquisition, and 
is a solid replacement for Randuins. If you find yourself struggling, don't be afraid to pick up 2 or 3 tanky items; your kit still lets you do crazy damage to isolated targets (just not as much). If you are doing REALLY well, you can substitute Randuins with 
Items i would stay away from are:
(zeal gives wasted stats, sheen damage not needed)
(the flat AD is nice, but honestly Hydra is just a better choice for him as it lets you 2 v 1 their squishies with the AoE burst)
(if you need the armor, get Frozen Heart; this item is just weaksauce for Kha)
(this item used to be a staple of mid-lane kha, who would spam his OP rockets then roam to other lanes to gank. Unfortunately, the missiles have been severely and repeatedly nerfed since then, and buying a tear will set your build super far behind.)
Other tips:
Wards are CRITICAL, don't just leave them to your support to buy. Get the upgraded yellow trinket, and always have wards. If you don't have vision, you will end up jumping onto what you thought was an isolated target only to find 2 more teammates there ready to squash you. Some good warding spots are the enemy wraith camp, the bush by the red camp, and the intersection of the wolf, blue buff, and gromp camps.
You are not J4 or Xin Zhao; your job is NOT to initiate fights. Wait for a few CDs to be blown, or for priority targets to be isolated, them jump on them and watch them poop their pants. This takes a lot of practice, and is something i still struggle with.
Don't be afraid to blow your ult to escape if things look dicey; the cooldown is really short.
You generally want to avoid picking Khazix if your team has two AD solo laners, but one laner he synergizes pretty well with is Talon. If you gank mid, you should be able to snowball that lane hard.
Junglers Kha struggles against: Sejaunia and Nautilus (they are too tanky for you to reliably kill on your own, and have the CC to kill you with their cinderhulk AoE), and Rengar (honestly, his window of opportunity comes up less often than Kha's, but Rengar can and absolutely will obliterate you if he gets the initiation.) Bear in mind that Kha is pretty dependent on his evolutions and items, so you should not usually be looking to fight the other jungler early to mid game.
If the enemy champs have escapes (Ez's blink, for example) try to wait for them to blow that before you jump on them (common sense, but easy to forget in the heat of battle)
If you have more questions, just let me know. Good Luck!
Exhaust/ignite Summoner Spell I use
so you can spam them.
against someone like 
, and even then you are probably better off just building 