This is just what I have worked out for jax and what works best for me so I decided to share it. I play a very very aggressive jax but it usually pays off. I have gotten more than one quadrakill with this build and one of them was literally a one vs four. With your jump you can jump past the tanks and to those nice squishy mages that hide in the back and take them out quickly with a stun and a few more hits and then work your way through their team and you are not squishy thanks to your passive and a good bit of dodge to help you out with those dps people that can get very annoying. Jax works well for 3v3 or 5v5 and actually can classify as the tank in 3v3 thanks to his dodge and lifesteal/spellvamp.

I use the 21-9-0 because Jax is a very offensive character and the 21 just add to the damage he can output. The 9 just adds a little but to his survivability and gives him the extra dodge and the little speed boost that is enough to get in range for a jump and stun to finish off someone that is running away.

Many people go for the rageblade before the gunbalde but because of the gunblades active and its lifesteal and spellvamp it definitely is a more powerful item to have. The deathcap just gives you enough ap to dish out a good bit more damage and lichbane really is a powerful item once you have that much ap. Also late game the trinity force is normally substituted for another gunblade for the extra lifesteal. I start out with the armor and health pots so that I can harass while staying in lane for a while early on without having to worry about being at low health. After you get your boots next comes the lifesteal so that now that you are out of health potions you should still be able to stay in lane for quite a while before getting the spellvamp and ap to make you last even longer. The rageblade is a must on jax and his next item because this helps with your attack speed and again boost both ap and ad which are the two real things that jax needs since they both give him health also. Once you have your deathcap at that point in the game you should be able to 1v1 anyone because you will be dishing out so much damage.

Honestly flash and ignite are just a personal preference if you like ghost over flash go for it but I would recommend keeping ignite because between that and your ultimate's passive they will wonder where their health went when they started the fight so well. Flash is helpful because it makes the getaway much easier after the turret dive and who doesn't love that little jump though the wall when you are being chased to help with that getaway.

Your jump is going to be your main ability and you are going to deal out some heavy damage with it and it is a great early harass. Do not forget to use your empower as it just adds one the damage. and don't forget to spam that stun when you can. Jump can do a good bit of damage and but has a little bit too long of a cooldown which is why you need to up it early to get the cooldown time down. Also, you need your stun but the extra levels just add some damage and a little but more dodge chance but its not enough dodge chance to make a significant difference and it is better to have the damage in your empower as you will be using it a lot more, and of course up your ultimate whenever you can.

I look more at the late game in my runes and so I prefer the bonus at level 18 especially since by around level six you are getting better bonus that the straight bonus at the beginning runes. Also, every bit of dodge you get on jax makes him harder to kill and that much more powerful with his stun.

In lane unless you are laning vs someone that has alot of range they should be scared to be close to you from your harass. Just sit there and keep fighting the minions until they get close then empower jump to them and stun them and add a few hits and they will be staying back so that you can farm all you want. I would not recommend taking middle though. In 3v3 Taking top is very helpful because you can get the early level advantage get an even earlier dragon and jax is a great 3v3 ganker thanks to his jump they won't have time to run because as soon as you are mia you are already mid jump and likely getting an easy kill.

Pros
-You are a hybrid and not very easily countered.
-You do a lot of damage
-You can even solo baron towards the end of this build or for 3v3 you can solo dragon at around 7 depending on your items.
Cons
-You are still a little squishy and vulnerable to heavy cc teams
-You get focused often
-Your only cc is your stun which you have to get a dodge for first and your gunblade

Play aggressive you can tower dive rather easily early with a jump and then flash out over a wall and later on you steal so much hp from them with a jump or just during the fight you can tower dive anyone that isn't a tank below half health. Also, you should be able to win most 2v1s and even a few 3v1s because of your damage output and all the health you are gaining back from your lifesteal and spellvamp.