Hey,
This build is a build that I have tested out myself and found it really effective. This build allows you to take a lot of damage late game and allows you to solo the barin as soon as you get the atma's impaler. This build requires you to farm really well or get a lot of kills. As Jax you tend to be focused a lot due to your damage output and I have constantly found it frustrating when that happens. This build focuses on being able to withstand some damage, but also to take advantage of his passive. With this build you should be able to reach close to 5000 hp.

Well that's the intro for my guide!
Pls do vote on this guide and if you down vote, leave a comment saying why you did it please so that I may improve this.

Pros:
-High damage output
-Can adapt to many situations
-Possible to win in a 1v2 situation
-Great passive and abilities
-Dodge and stun may save your life when they decide to turret dive you.

Cons:
-Focused a lot
-Is harassed easily by ranged carries
-Target of many CC abilities.

The reason I get those runes is because they have the potential to max out your level 1 leap strike damage and possibly obtain first blood. The dodge chance might not seem like much but who knows when it will save your life when you dodge a shot under your turret and score a kill when your opponent decides to tower dive you.

You get boots and health pots first because it gives you the needed speed boost to either chase or run away after first blood. The health pots are to help keep you in lane longer and hopefully able to sustain. You until you can get the ninja tabi.
I get pickaxe for Guinsoo first because it will raise the damage of your leap strike by giving it extra damage. You grab a Guinsoo because it synchs really well with your ult. This item will also let you push turrets faster and also will increase your health pool when you attack due to the passive. =D
The hex tech gun blade is there to nicely give you both the spell vamp and life steal that Jax needs. This item also synchs well with the Guinsoo as it is giving you the same amount of hp you are earning, even if only temporarily.
Now the warmogs many of you may think it's useless but it is one of the most useful thing on Jax. It gives you the health you will need when you jump into battle and also will allow you to be an initiator if need be, although I highly do not recommend this.
Now one of the main reason I buy warmogs is also for this item, Atma's Impaler. This item is an amazing item with Jax since with your high hp pool it will give you a decent amount of attack damage, but also will give you more health because of your passive. Now if im not mistaken your passive and Atma's will stack on top of each other which is pretty amazing.
Now for the last item you don't always have to get a Rylai's but instead you can get a Trinity force instead if you really want it, I have tested this out and it does work really well but I prefer to reach 5000hp and have another slow because sometimes chasing Singed (I know it's not smart but I do it sometimes because he's the last one left or he has really low health and I have over three quarters health left) is really annoying because of his high hp pool and your hex tech active only lasts for a couple of seconds.

There are many summoner spells that you could use but I personally prefer flash and ignite. There are many chances of escape when you flash over a wall or use a empowered leap strike to turret dive, ignite, then flash back out. If you really want to then you can use your own set of summoner spells but please don't take something like rally. It might help you early game but it isn't that helpful late game and the cool down is just ridiculous.

The reason why I max out leap strike first is so that I can do a lot of damage instantly but also to lower the cool down so I can use it more often to escape or chase. Counter strike is next because it raises your dodge chance and will let you stun more, because in team fights your aoe stun could mean the difference between victory and defeat.
Level up your ult whenever you can and activate it when you are fighting casters or jumping into team fights. Hell, sometimes I use it when going in to fight the other champs in lane in order to reduce as much damage as possible.
When using leap strike always activate your empower because it will add extra damage to your jump and also sometimes may allow you to kill champs faster.

That's all for the guide right now. I'll right up how to solo baron later if you need it. Post in the comments if you have some suggestions and sorry for writing so little and no pictures. I can't be bothered to put anything pictury in the guide cuz I'm usin an iPad to write this up and like I said , I'm too lazy. I'll put up stats of some of my games with Jax later. I'll try and update this guide once a week.

I hope this guide helps you in playing Jax as he is a very fun champ to play.