Not Updated For Current Season
Jax has been nerfed several times and he is about to lose his op status. So one might consider a good build now ^^
First things first:
Not too surprising:
Attack Speed to get 2.5 Attack Speed with ultimate lvl 3 and the 2 main items.
Dodge because you are Jax.
Cooldown Reduction to get the Maximum of 40% with Nashors Tooth and Mastery.
9 Points in defensive mastery are a must have for Jax to get Nimbleness.
21 Points in utility tree are often dismissed for offensive mastery but i played both pretty often and consider utility tree more utile ^^
#1 you get more Cooldown Reduction
#2 you get movement speed increase
#3 you get more mana regeneration, more experience, less dead time and longer neutral buffs if you want.
#4 less cooldown on summoner spells
#1 less physical damage (everything leads to more damage in offensive tree, more criticals, stronger criticals, more attack speed, more damage...)
#2 you dont get the 15% magic penetration (also damage increase)
#3 you dont get your ignite give you 10 AP when on cd
There can be an endless discussion whether you take offensive or utility tree. Some also consider defensive tree.
I chose Utility because it is definitely the strongest tree on low lvl (1-5). The mana regen makes you mobile and destructive. Lower cooldowns on ghost and ignite often gives you one extra kill. A Jax with utility is not as predictable as the other ones that's why i favor it.
In the beginning doran's shield is the best item. You get HP and you get HP regen. That's all you need. Combined with a potion and lvl 6 can be reached easiliy without recalling.
After that boots of speed are the next item. Melee champs need to be fast. No questions asked. You can make them Ninja Tabis for Dodge and some Armor. Sometimes it is better to get Doran's Blade instead. This is when you face Ryze and Fiddle on your lane. Or you face Janna and Morgana. You don't need Dodge then you need HP.
You can get up to 3 Doran's Blades. Many guys will call you noobs but they have no idea. 4 Doran's Swords are the same as a Pick Axe, a better Giant Belt and a better Vampiric Scepter. And it is cheaper by 860 Gold. The whole thing is called Doran's Blade Massacre as invented by Freundschaft. Don't underestimate it. Gives you great tower dive opportunities on lvl 10.
But in most cases you run best having Doran's Shield, Ninja Tabi, Doran's Blade and then you go for you main items.
Main Items: Nashors Tooth and Guinsoo's Rageblade. These 2 Items provide everything you need. Your Cooldown is reduced by 25% + 6% Mastery + 5,85% Blue Runes + 2,34% Quintessence. Almost 40%. You get Leap Strike every 3.01 seconds. Attack speed is enough to get 2.5 Attacks per second if you got 8 Stacks on Guinsoo's and 10 Stacks on your Ultimate lvl 3. If you want to be a killing machine you need reduced Cooldowns and max attack speed. These two fit perfect for Jax since he really needs attack speed and benefits from AP.
At this point you have Doran's Shield, Ninja Tabi, Guinsoo's, Nashor Tooth and 0-2 Doran's Blades. From this point you are a killing machine. In most cases there is no champion who can kill you in 1on1. 1s stun with counter strike every 3 seconds, leap strike to chase down everyone with 3 second intervals and crazy damage with your ultimate.
After that you need to analyze the team fight dynamics. Banshee's Veil is in most cases a the best decision. Hit Points, Magic Resist and neutralize a spell every 30 seconds. On the other hand a Bloodthirster gives up to 100 damage and 25% life leech. 100 damage means 300 HP. But if there are no mages troubling forget banshee's veil. Maybe get even get a phantom dancer for more dodge and critical which is good vs. physical dps champs. If you decide on phantom dancer you can sell your guinsoo's. Frozen Mallet or Rylai's are also considerable. For Jax both are pretty close to each other. Just be flexible. If you switch to phantom dancer you can also get infitiy edge. Then you can also exchange a bloodthirster for an executioner's calling so you get >70% critical strike. But if you just get nuked all the time a Guardian Angel can still be the best item for you. I'm sure you got the point.
- You are no tank. But you can initiate team fights. After that you run out of it and if enemies chase you they die.
- In early game just leap strike your enemy and run back instantly. Many opponents will also run away if you leap strike because they expect you to follow. This gives you free hits without counter strikes. Add empower of course if you have enough mana.
- Remember your enemies Summoner Spells. Don't chase a guy with exhaust. When his allies come up you get exhausted and die if you have no ally to jump to. Use ignite against enemy life leechers. It gives you a great advantage.
- Against enemy teams with 4-5 mages get mercury treads and banshees veil. It is worth it.
- If you escape a team fight don't move too far away. Your cooldowns are low and you can change battles just with empower/leap strike even if you are low on hp. Run out of combat after it. Gives you kills. Drives enemies crazy very often and supports your team.
- Get red buff. It suits perfect for you. No need of Rylai or Mallet if you got it.
- You can kill many minions with your ultimate and empower (and counterstrike ofc). Make use of it. If the opponent team plays conservatively just get your gold by creeping. You are a good pusher dont gank too much.
- Slay them all - or in LoL-language: W-Q..proc ultimate...E, W, proc ultimate, Q, +300 Gold thx and imagine if Jax had a real weapon!