Jax is, for the most part, the most pivotal character of any Dominion line-up. He is often powerful enough to go up against any single opponent one-on-one, and even mowing down enemy Jax users. Everything that he does can turn the tide of any combat whether it's crowd control (though limited), singling out an opponent in a group and finishing them quickly, or even, and most importantly, diving into the range of Dominion turrets to initiate and even finish combat before capturing.
Unlike most of the other 'hybrid's you see, this build focuses on the AD aspect of the hybrid, only putting in just enough to get what's necessary, while other Jax builds would focus instead upon Rylai's Crystal Scepter or go for the all around special effects of Trinity Force.
This build instead focuses on the direct hand to hand combat capabilities and increasing the effectiveness of the already more than prominent skills that Jax possesses. With this build, you'll be able to handle even two verse one combat scenarios should the need arise, as well as feed on minions when you need to heal.
Jax the Armsmaster. May the best man win.
Obviously the most pivotal of all guides, the items. The first step is of course, the Tabi Boots for the Counter Attack on top of survivability. Your dodge will make you a foe to be reckoned with in equal combat. Then again, any experienced Jax user knows this well, so on to the next item.
Guinsoo's Rageblade. Dominion is extremely quick to level six, so the minute you get this you're more than likely getting your ultimate, Relentless Assault. Aside from the obvious of the overall boost in every stat (except for dodge, defense, and resist), you're also making it so that the moment you start combat, you're already getting the boost from the blade, and in scenarios where your opponent begins to run, you can use your leap strike to keep from losing your entire speed boost (as well as getting a little extra damage since Rageblade also increases your AP.) Overall, this is the best buy for a up and coming Jax.
Next, you'll want the Hextech Gunblade. But before you get the Hextech itself, you'll want to focus on the parts. First thing's first, if you don't already have enough to just buy it straight out, you'll want the Vampiric Scepter. This will guarantee that your survivability in straight up combat jumps, especially with your ult and rageblade's speed boost. Next, get the Hextech Revolver, again, this will boost your survivability by leaps and bounds since all of your skills will feed you much needed HP. Afterwards, the rest is elementary. The Hextech Gunblade is extremely effective in returning your HP with all of your attacks, AD and AP based. When it comes down to it, at this point you're feeding yourself enough on all attacks to force most opponents to back down.
Now, for where most guides flop around...
Sanguine's Blade. By far the most powerful weapon in any AD fighter's arsenal. All of Jax's skills with the exception of counterattack hit as though they're your basic attacks, triggering Sanguine's increase in damage and life steal. Far more effective than most people give it credit since Jax himself is focused largely in quick combat, often laying three or four hits in the time it takes opponents to lay one or two. On top of just having a large increase in damage, it also increases your life leech on attacks, giving you that much more survivability in a one on one (as well as two or three sometimes) fight with opponents.
Often, by this point, your team or your enemy's team have won. If they have not, however, your focus should become that of on hit effects. I usually focus on things that I can use by myself that would target those who would try to combat me on my level. The specific focus should be on what is necessary for your particular situation. Typically, Entropy Hammer is a must in all situations to fight against people who run, however the previous weapon is variable. I personally prefer the Axe because I can fight people who try to combat me on my level as well as adding even more damage on top of an already hefty basic attack, however Wit's End and other weapons could be used in it's place depending upon what you have to fight.
Obviously, the set-up of skills from early on verse later on is a bit of a difference. The first thing you want to do is get one of each skill. This is the most pivotal thing, as any attempt at fighting requires all of your skills, even if they aren't upgraded.
After getting each skill, you want to focus on Leap Strike (obviously maxxing your ult every chance you get since it's your meal ticket.) The reason being that your greatest asset is the ability to literally leap into combat, killing the range advantage of opponents and more importantly, leaping through the range of turrets that you'd be soaking up damage from the turret and the opposing champions. Maxxing it is especially important because the more times you can leap, the more times you can get shorten the range between you and your victim who will be trying to max the damage you take from the turret.
Afterwards, you'll want to max Empower. Not only does it deal extreme damage by itself, it can also stack with your Leap to max the damage of the inital strike, as well as start up the cooldown for your next Empower a little early.
The final skill, though most pivotal in later fights, is your counter strike. Every time you see it ready, you focus on how you want to use it. If you're in the middle of a major fight, you be ready to hit your counter strike immediately. Regardless of the fights you're in, those build ups of 1 second are extremely useful, especially in cluster fights with multiple opponents. If you manage to get one from a minion, and you have the spare life, it's a good idea to use an Empowered Leap Strike and hit your counter immediately to stop your target from running (even if they're under a tower.)
As for your ultimate. Cap it every single chance you get. While it's effect, when read, doesn't seem like much, it's increasing speed as well as every three strike damage is instrumental in making Guinsoo's build as powerful as it can get as well as letting you take advantage of the AP and AD leech and on hit effects of your late game weapons. On top of this, your active is a great way to counter mages who give you trouble even up close, bolstering your resistance enough to make most opponent's great attacks drop in power too vastly to even put a dent in you.
As far as spells go, you can mix and match as you feel necessary, though a cleanse/exhaust combo have tasted most delicious when facing opponents who like to slow and stun.
Revive, Exhaust, Ignite, and Cleanse have been the most effective in my experience. Being Dominion you're basically going to die at least once or twice, so regardless of the situation Revive is a nice investment, especially if you decide to be a more offensive type.
As for Promotion and Garrison, they're also good options if you decide to play more conservatively. You can use promotion to make a super minion for the turrets to chew on while you dive, or use Garrison to make the turret's damage weaker (as well as restoring your own if you're on the defensive.)
Pros / Cons
Great survivability in one on one combat (especially with the right spells)
Ability to soak up damage while fighting.
Extremely good turret diving abilities when necessary to take towers.
Can fight all known AD fighters one on one with a high chance of victory due to AP and AD Vamp.
Susceptible to Crowd Control if cleanse is in cooldown or not equipped.
Opponents who can break chains (Ex: Vlad and Ryze) are difficult to fight over long periods.
In short, this build specifically is built towards Jax's stregnth of speed and power rather than his ability's power and higher HP. It's a build that can defeat even the strongest of tanks while maintaining the burst power and speed to defeat the mages who would harass melee types. It's served me faithfully keeping me in the top two spots of every team I've used him in and forced opponents to either back down or gang up to beat me, which even then doesn't always work.