Hello!

This is the build I normally play on Jax, and while I'm not doing a full guide, there are a few pointers I want to show:

You don't have much sustain. None of your items give you health. YOU ARE COMPLETELY FOCUSED AROUND LIFE STEAL. This is very important. If you can't get the life steal, then you're not going to be able to survive. If you don't get your attack speed, then your life steal won't work as well. However, if you manage to get the full build, you can solo Baron Nashor and barely lose 500HP.

Please leave feedback once you try it out and let me know how it worked for you.
Good luck, and have fun!

There is a problem with this build however: Crowd Control. As I noted before, you have absolutely 0 sustain. If you are stunned, caged, or slowed, you won't be dealing damage, and therefore, you won't be gaining life. Once you stop gaining life, you stop living.
This build will not work against champions with heavy CC ( Leona, Nautilus) or a team comp with heavy CC (i.e., Ashe, Ryze, and Cho'Gath all on the same team).

Another issue is your weakness early on. You can't do much in lane to start with, as you don't have much to work with. You don't have sustain, and can be poked easily. However, I usually find that as things start to change for the better as soon as I pick up the Blade of the Ruined King.

Runes:
I would advise Health, Armor, and Magic Resist runes. You shouldn't need to much in the way of Armor Penetration or extra Damage, but having sustain runes can be a great help. Or, if you just don't care about sustain, then Armor Pen, AD, and Movement speed would all be good ideas.

A few items can be switched around or swapped out.

Nashor's Tooth
Hextech Gunblade/ Zephyr can be swapped out for a Nashor's Tooth. Swap this with the Gunblade for more AS, or swap it with the Zephyr for more AP. Building the Fiendish Codex or the Stinger first would, once again, depend on if you needed AS or AP more urgently.

Mercury's Treads/ Ninja Tabi
This would depend on who you're up against. While the Berserker's Greaves will give you extra AS (this build + passive = 2.5AS), they don't give you anything in the way of defense. If the enemy is heavy physical damage, then you'll want to build Ninja Tabi. If they have a large amount of CC or are mostly AP, then you'll need the Mercury's Treads.
Enchantments
If you have the time/money, either Alacrity or Furor will be good, it would just depend on if you want burst movement speed or flat extra movement speed.

Frozen Mallet
While you could replace the Zephyr with this, this item would fit in better if you build it instead of the Hextech Gunblade and build a Nashor's Tooth instead of a Zephyr. One could also go with a pure AD build and build it in place of the Gunblade, keeping the Zephyr.

Maw of Malmortius
This item would fit in same as the Frozen Mallet, but while the Frozen Mallet is good against physical damage champions, the Maw of Malmortius would be of use only against heavy AP champions.