Hi guys, an important note here.
The guide itself is coming up. I'll leave it at the item build and a few clarifications. I published it already because I want to know what you think about this build, after trying it.

This guide was written for summoner's rift.

Max Leap Strike first because of the lower cooldown. This skill lets you jump where you please and does great damage if you hit an enemy with it. You can target both allies and enemies, so when you're attacking, be sure to click the enemy champion and not your minion standing beside him. The low cooldown on lvl 5 makes chasing and escaping a breeze.

Max Empower second to max out the damage on your Q+W combo. You'll make squishies wet their pants. I don't max this first because, even though it does more damage than leap strike, you're not very likely to always hit the enemy with it every time it comes off cooldown, unlike leap strike. The extra damage from lvl 1 empower is great anyway, because the gunblade adds a great amount of damage to it.

Max Counter Strike last because the cooldown is low at level 1, the damage it does won't get much higher by upgrading it and you really don't need more dodge chance.

Get a point in Relentless Assault whenever you can, you'll notice a huge increase in damage output whenever you upgrade it. This skill provides you with all the attack speed you're going to need.

You might wonder why I chose these items and didn't choose others.

Why no Doran's Shield? Because, frankly, I think the health regen isn't noticable and any good player can zone a Jax out easily, even if Jax has a shield. Therefore, I'd rather get an Amplifying Tome to start with, remaining in the bushes and jump/run out from time to time to last hit a nearby minion or to burst an enemy down to half health with the Q+W combo, killing him easily when he comes too close.

NOTE: Getting mana potions is THE way to prevent mana issues. Buy them. I usually buy about 6 in a game.

After finishing the Hextech Revolver, you'll have great spell vamp, which lets you get back to full health in seconds, once you have your ultimate and when you use W a lot.

Why don't I get Ninja Tabi? Because Jax has more than enough dodge from Counter Strike and the dodge runes, more dodge would be overkill and can easily be replaced by other boots. I choose to get Mercury's Treads mainly because of tenacity and partly for the added magic resistance. I am aware of the fact that the tabi also grant magic res because of your ultimate, but it's less and it's not as reliable as mercury's. Most of all, Tabi's don't provide tenacity and I don't think any of the new tenacity items are fit for Jax.

Why do I get both a Hextech Gunblade and a second Bilgewater Cutlass? Because of the insane damage and lifesteal+spellvamp (solo nashor ftw), for the two slows and a combined nuke of 500 magic damage (the two actives aren't identical and therefore you can use them both). The two slows combined will provide you with a combined slow duration of 6 seconds, which is great for Jax, because you won't need to waste 3k on a Frozen Mallet.

Why don't I get Guinsoo's first? Because a revolver allows you to do one important thing better: survive. for only 1200 gold, it lets you kill every creature in your or the enemy's jungle without losing health. However, you should only do this once you got your ulti. Upgrade it to a gunblade and you'll be soloing dragon in no time. The rageblade is more cost-effective if it comes down to pure damage, but the lack of a slow and lifesteal/spellvamp are good enough reasons for me to get it AFTER the gunblade and the cutlass.

There's no reason for not getting a Guinsoo's Rageblade. It only costs 2.2k gold, it provides you with 35 AD, 100 AP and 32% AS when fully stacked and this translates into 305 EXTRA health because of your passive Relentless Assault.

Why do I get an Atma's Impaler? Because it works great alongside your passive. The items you got before this one give you 735 bonus health. Together with you being lvl 18 (which you probably will be, around this point), you now have 2784 health, which means your Atma's impaler would give you an extra 55 AD. However, Atma's stacks with itself ONCE, so it gives you another 165 health and another 3 AD. 45 armor, 18% crit chance, 58 AD and moar hp? Yes please.

Jax has great AP Ratio's, so getting a Rabadon's Deathcap would be logical. This lovely item increases you AP by about 250, it makes your skills hurt a lot more and it gives you 500 health.

Against a competitive team, there's a good chance you'll build all these items. Make sure to farm a lot and gank every lonely straggler you see. I can't think of a champion who is a better 1 v 1 than a Jax using this build or a more tankier one. People won't be able to escape from you because of the dual slow and the short cooldown on Leap Strike.

If you think you die too much and you want to get more survivability, steer clear of items that give you more health (like sunfire cape or warmog's). It's a waste of money, because you'll have soooo much lifesteal/spellvamp with just the gunblade and the bilgewater cutlass.
Get a Chainmail and/or a Negatron Cloak first and upgrade it ONLY if you feel you ABSOLUTELY need to. When you decide to upgrade them, I recommend getting a Warden's Mail and/or a Quicksilver Sash, they provide awesome utility for their prices.