This is my first guide so dont be angry about small mistakes and it's only how I play Jax and how I think. Also I know that most of persons hate Ionic spark item but on Jax he works really well. Try it if you want.

On jax almost every summoner works well but the best are:
Ghost- very usefull tool on jax becouse you are more dps than burst champion and you will need some movement speed.
Flash- Not much to say just more mobility. Take it if you are new with jax or/and you don't take ward with yourself.
Ignite- For me its just a little bit of extra damage and finisher. If you need more damage than take it for sure
Teleport- you will need a lot of farm so teleport can help you to farm yourself and don't miss any teamfight.
Cleans- If enemy has Skarner and Warwick or Malzahar and youre team doesn't have any true tank so you need to initiate teamfight take it.

Most of you will ask "why ionic spark" my anserw is- with ult you will do insane damage and it will give you also attack speed and health which are really important on Jax. If you dont want it you should build Fantom Dancer but only after Frozen Mallet. Also if youre losing lane and need sustain build Wriggle's Lantern seling afterwards to get bloodthrister or Hextech Gunblade.

Why I max q first? For better damage and larger mowement options. If you want you can max w instead but it makes you less efective in lane or ganks if you will push enemy tower. Maxing e is compleatly worthless so please dont do this becouse you will just fail.

Pros
Great 1vs1 fighter
Strong burst
Nice harras
Hard to counter with items and champions
Propably best champ in late game EVER!
Strong jungler and ganker

Cons
No sustain early
Countered by Singed ultra hard
Exshoust makes you usless for entire duration

As I said it's my first guide I hope you enjoyed it. All wirtten by me. And sorry for no runes but there is so many choices (i use half ad half ap). Have good day and dont forget to comment and vote :)