I have found Jax to be one of the best champions on the Twisted Treeline and almost impossible to counter. While I am not the best player on 3s, I do feel that I have a pretty good system for playing Jax on it. Enjoy my in depth look into Jax.

PROS:

-Lategame GOD
-With the combo of Counter Strike and Leap Strike level 2 kills are easy
-AOE Stun
-One of the best duelists in the game
-Hard to kill once you have Hextech Gunblade
-Really good farmer/Tower pusher
-Can work alone


CONS:
-People hate you. You will be focused.
-When abilities are on CD you can be very vulnerable to burst.
-Kiteable
-Nothing else really... Just bad teammates

SEALS
I find that flat armor is the best for early game fights.
GLYPHS
MR/level is the way too go because most magic damage comes lategame.
MARKS
I go AD/level. But I think many different marks could work on Jax. I just find that the +35 AD at 18 is like having another AD item. And considering Jax is a lategame champ it really helps. Also as the AS bonus from your passive levels up it compounds it with more AD per level.
QUINTS
I go AD/level also in order to get the +35 AD at level 18.

The offensive tree is simple to fill out for hybrid Jax. But the wild card I choose is Phasewalker because it allows you to recall faster especially when you splitpush down a tower and get stranded from your team.

*Example- Many times a 2v3 emerges bot while I'm in jungle. I use this to push hard top using all my abilities. My team can turtle under tower while I get a few hits on turret. I can either draw a fight out of one of them who heads top. But if all 3 leave bot lane I just instantly recall right under their turret while the minions chip away and they get no kills vs. the turret dmg I dealt.

Allows you to duel anyone top lane to secure first blood. Also it shuts down carrys or can stop a fleeing enemy in their tracks.
Ignite allows you to secure kills mainly that first blood top. I also find it uselful for dueling people with sustains like lifesteal.

This is what allows Jax to build high AD/AP without worrying about AS. Makes him a monster duelist and tower pusher.
This is a great escape. But I normally use it to set up a stun from Counter Strike.
Great AP punch. I use it early for harass and last hit CS. It allows you to optimize your dps if used between basic attacks.
Your main ability, more valuable then your ultimate's active. Get this at level one if the enemy team trys to raid. Stun them all and focus the squish. First Blood.
Pop the active as soon as your opponent commits to a fight, it will make him rethink the fight. The passive 3rd strike it what makes you a hybrid, so good.

THE COMBO
If you sit in a bush and an enemy gets in Leap Strike range, start your Counter Strike and then Leap Strike to them. You will at least get some major harass in.

Best Case Outcomes:
-You get a kill
-You make them waste Flash (a counterable spell imo)
-You make the opponent back (Tower damage/ Free CS/ XP advantage)

The more of these combos (starting at level 2) you can pull off top lane the better chance you have of snowballing.

You won't need more the 2 health pots before you can recall to upgrade to a Vampiric Scepter. But I do not start with Vampiric Scepter because Boots are too valuable at level 1.
Next you want to upgrade your Vampiric Scepter for more sustain and the active give you a ranged slow ability which is quite useful.
I normally try to get this next as I rush Hextech Gunblade. If I recall and don't have enough for it I will often buy an Amplifying Tome or two.
**CORE** This is the most important item for Jax it give him so much sustain and dps at the same time. Also the active improves his CC and burst potential.
**CORE** Depending on the Team Comp and my current gold at the shop after purchasing Bilgewater Cutlass, I will buy these because I don't need to get Ninja Tabi, Mercury's Treads or any other boots thanks to Moonflair Spellblade's Tenacity and the Armor/MR it has. The Speed also makes Jax less kiteable.
I also often buy this item after Bilgewater Cutlass if the opposing team is AD heavy as it leads into Moonflair Spellblade. Alternatively, I can also buy a Negatron Cloak if the opposing team is Magic Dmg heavy as it too builds into Moonflair Spellblade.
**CORE** This item does it all. It give you Armor and Magic Resist while giving you some AP and the tenacity is huge because opposing teams with try to CC the hell out of Jax.

^^THE ABOVE IS THE CORE OF YOUR BUILD^^
Anything from here on out doesn't need to be built in any particular order
An item made for Jax. AD/AP/AS all at a good price. The "Frenzy" passive is also great in a duel.
TONS OF DAMAGE. Any hybrid's best friend.

SITUATIONALS

Nice CDR, AS, and AP. The unique on hit passive is great lategame for your dps.
TONS OF HEALTH. You have the resistance and damage, now get a little beefy.
Rarely used. AP is useful. Has good Armor. I only use for the active on teams that have dunk ults like Garen's Demacian Justice or Darius's Noxian Guillotine.