First I'd like to say some words but I wont drag it since I know you guys just care for the build, but please read some parts like the item choices because it will make you understand the champion a bit more. You are a good player of you got the feel of a champ.

Anyways, this is my first guide, bla bla, though I hope you wont underestimate my experience just because I'm lazy to write guides xD Also I'm not saying my guide is the best or even perfect, but what I can say is that it is built around logic and not "it worked good for me the last games I played". Again, ALSO, enjoy this guide ;)

Even though the OP champion known as Jax was nerfed since the removal of Dodge, he still lives up to the title of "Grandmaster". His Ulti was also recently nerfed/buffed into something defensive(luckily he still got the passive :S). In my opinion, it's great because now there is more reason to still build him a bit tanky and with further sustain while still being able to pull of crazy damage.

Who is Jax?

He is the Grandmaster of course :D Aswell, as one of the most bursty damage dealer's in the game which excells at picking of single target's one by one. Going 1v1 on Jax? Good Luck! A fed Jax can take on a whole team with breaze. Jax can also play almost any role in a game though I'd say he excells mostly at Jungling and Solo-Toping.
He makes a great Solo-Toper because of his ability on dueling champions 1v1 and harassing.
He makes a great Jungler because of his Jungle speed(passive stacks AS and E is still a Dodge but worse) and because he is a natural ganker(one of the best I would say because of he can burst, stun and dash).

Why my build?

Just because you know you want to :D And maybe also because as I said before, this is a build that utilizes his abbilities and synergiezes well with his style. For example, with Trinity Force, you get to slow the **** out of them and get that basic attack buff each time you spam your abbilities(especially W, which becomes availble every 3 seconds). So it gives you mad bonuses. It makes a great item for a Tanky DPS Hybrid.
Lich Bane is the choice for my AP based Hybrid Burster build but it is not built as early since you will still want to have a decsent AD to have you going with sustain in early/mid and also be easy on your mana. So obviously, the Burst build is way more effective late. That is why i would RECOMMEND the Tanky DPS for a great a start and a great late rather than a weak start and a crazy late because you might not even get so far.

+ Great burster. - Reasonably bad with CCs.
+ Does great damage even without - Stun can be very predictable.
that many damage items. - Without Offtank, Jax becomes
+ An amazing ganker. very squishy.
+ Has high speed and great sustain in - Has reasonably high mana costs
jungle. early on.
+ Q can dash to allies within range.
+ One of the best 1v1 champ.

I have played Jax in many different ways and I found that an AP based Hybrid Burster and a Tanky DPS Hybrid works the best and utilizes Jax abbilities and stats the best.
As the burster you are going to deal sick damage which is used to burst the target in a few seconds but if you become a very easy target and VERY vulnarable for CCs and ganks.
As the Tanky DPS you will still deal almost identical DPS but also have the ability to chase