Hello guys,my name is Arion,i am 17 years of age and I am Greek( So,sorry for any grammar mistakes made in this guide :P ).
So,as you have already figured out this is a guide on Jax.Jax is my main,I have over 200k points on him.Only playing top for over half a year now,so i think this is enough to make a reliable and good guide on him!

For runes you will need flat ad reds for the early trades especially lvl 1-3.
And a red crit in case you are feeling lucky and want to surprise your opponent with a 1/50 chance double damage auto :)
For the seals i run 8 flat armors and 1 hp.
Now on glyphs I personally tend to get flat MRrs for early trades as well,changing to scaling per level MRs if my opponent is ad.

For masteries its standard,except that i go 3-2 and 4-1 in the ferocity tree.Thats cause you will need a bit more ap,but less ap penetration because most players will focus on getting armor,plus you ad is 3-2 times more than your ap!

Ok,this is a tricky one.It heavily depends on your playstyle.
With jax you want to splitpush a lot.Getting the wrong one can lose your game before it even begins.
-Ignite is a must,because you want to have that level 1 dominance with jax,and the possibility for a first blood on levels 1-3.
-If you are feeling getting fancy on the rift you should go with flash.Not only it allows you to make surprising "e-q-w-flash-e" combos,but you will get out of twice as much ganks as if you had tp.
-Tp is the safer choice for late game,as you can splipush without worrying your team dies alone in a 4v5.
-Ghost is the one I hate playing against because its good in the right hands,but I can't get used to it so its a no for me :P

Pros+
LvL 1-3 trades powerhouse
The best 1v1'er in the game,late game
Amazing damage while having nice sustain too
A god versus aa based champions,such as aatrox,or xin.
Easy to play,quite easy to master(to 90%)
He snowballs like no other top laner if fed.
Amazing splitpusher

Cons-
Useless if behind
Need to get used to his ignite-based early game
Very hard to reach his 100% potential,which you can get by getting used to his flashy plays,getting all the aa's without cancelling them,escaping with wards the right times etc.
Needs a lot of cs,early,so he can catch up with everyone else and slowball so he can carry in 2v2's or 3v3's!

You will either win teamfights by either landing a good "e" (3-5 people),dealing tons of damage,instakilling their backline,or taking damage while stunning the frontline and dealing damage to their offtanks.
Its very easy to teamfight,compared to other champs.The most important thing tho that you must keep in mind is that your damage will be doubled,if you get in the fight stacked(Your passive).A fully stacked late game jax getting in the backline with a helicopter "e" is a win.

You will need to improvise,as every jax has his own ways to win his lane,but there are some basics.
You hit the 3 fighter creeps 6 times,2 each,so you get your passive stacked,you hit 1 aa on the enemy champ,and when the caster creeps start making their aa animation,you open e and go all in on the enemy top laner.That will get your "E" dmg bonus,plus all of your aa's.After that there are 2 options.If they don't fall back as soon as they see you open your "E",and they fight after you stun them,you pop ignite and kill them.If they manage to get under their tower,with 1-200 hp,you go all in helicoptermode when you get your level 2,getting close to them,open e,q them-ignite them,do 1 aa and there there are 2 options.Either they flash,so you flash with your e open and you stun them(bad scenario),or they stay and get pwned by your superior skills,and you get out with a flash like a boss,getting hyped to carry the game.Oh,and I almost forgot,if they are over 200 hp,you just pressure them and casually ward the brush now and then for the possibility for a gank.

As Jax you should be farming for your first item,while having >1 kills in your record then roaming and start farming again.Your job in mid - late is to splitpush while your allies are taking objectives,other towers with low hp,barons,drakes,or elder drakes,so that enemies are attracted to you and you possibly 1v1,2 or even 3 them,remember,you have the ignite after all.

So,I am looking forward on expanding my guide's content in the next few days-weeks,so it becomes a very detailed,enjoyable to read and educational one!Every single day for the next week or so there is going to be an addition to my Jax-guide!

Thank you for reading till the end,hope you have learned something,and the most important thing of all,have a nice day my friend!
Arion-10-21-16