Hi, this is my first build and I hope you will like it. The reason why I picked Jax is because he is my main. My division is GOLD V and I played few games with this build and I won.

I had one more game but I can't find it. Score was 22/4/17.

Summoner Spells

Ghost

The best escape spell ANY character can have as well, 2nd to none to flash. Use this to either escape from a battle or quickly regain your position on the map, such as on Summoners rift, going to the aid of a teammate in distress/tower in distress of an enemy attack.

Smite

Smite is necessary to secure monster objectives such as Dragon and Baron and runic buff objectives such as Blue and Red. It also increases your jungling speed. A good thing to do when using Smite is to hover over the icon to see how much damage Smite will do and last hit the monster objectives once their HP falls below your current smite damage.

Flash

Flash is the best option for most champions. It works as an engaging tool, escaping tool, etc. You can use it to extend the range of your engage, or to get out of a bad situation after you went in with Leap Strike.

Relentless Assault

Jax's consecutive basic attacks continuously increase his Attack Speed.

Leap Strike

Cost: 65 Mana
Range: 700

Jax leaps toward a unit. If they are an enemy, he strikes them with his weapon.

Jax leaps to a target unit, dealing 70/110/150/190/230 (+100% bonus Attack Damage) (+60% Ability Power) physical damage if it is an enemy.


Empower

Cost: 30 Mana
Range: Self

Jax charges his weapon with energy, causing his next attack to deal additional damage.

Jax charges his weapon with energy, causing his next basic attack or Leap Strike to deal an additional 40/75/110/145/180 (+60% Ability Power) magic damage.


Counter Strike

Cost: 70/75/80/85/90 Mana
Range: Self

Jax's combat prowess allows him to dodge all incoming attacks for a short duration and then quickly counterattack, stunning all surrounding enemies.

Jax enters a defensive stance for up to 2 seconds, dodging all incoming basic attacks and taking 25% less damage from area effect abilities.After 2 seconds or if activated again, Jax stuns surrounding enemies for 1 second and deals 50/75/100/125/150 (+50% bonus Attack Damage) physical damage to them.
Counter Strike deals 20% more damage for each attack Jax dodged (max: 100% increased damage).


Grandmaster's Might

Cost: 100 Mana
Range: Self

Every third consecutive attack deals additional magic damage. Additionally, Jax can activate this ability to strengthen his resolve, increasing his Armor and Magic Resist for a short duration.

Passive: Every third consecutive strike Jax deals 100/160/220 (+70% Ability Power) additional magic damage.
Active: Jax strengthens his resolve, granting him Armor and Magic Resist for 8 seconds.Armor bonus is equal to 25/35/45 + 30% bonus Attack Damage.Magic Resist bonus is equal to 25/35/45 + 20% Ability Power.

Quints

Quints of Life Steal are the best choice beacuse they give you almost full health throughout jungle so you dont need to recall if you have low health.

Glyphs

Glyphs of Magic Resist I recommend this for early game, and for ganking AP champions.

Seals
Seal of Armor This is pretty much the standard for any jungler and there's real alternative.

Marks

Mark of Attack Damage Attack damage marks increase the damage counter strike and leap do. This increases your early game burst and helps you secure much more reliable ganks.

Mark of Attack Speed This runes are the best for jungling, it helps you deal more DPS to monters and clear camps more easily.

Trinkets

You want to buy this trinket first because it's good in early game. If someone is chasing you just put ward and jump on it.


Later, when you decide that you don't need Warding Totem sawp it to Sweeping Lens because it's good when you gank and they have ward you just use your Sweeping Lens to destroy ward and come back later.

Boots

First you want to buy these boots because they allows you to provide more ganks.
vs
At late game you will not need Boots of Mobility. You have to choose between Ninja Tabi and Mercury Treads. It's simple, if they have heavy AD team you wan't Ninja tabi, if they have heavy AP team you want Mercury Treads.

Feral Flare

Feral Flare is what lets you clear the jungle and will be one of your main sources of damage.The item was released to give carry junglers an offensive jungle option. Wriggle's Lantern was reworked and made much better but it was also given the Feral Flare evolution which made the item have actual late game potential. It granted a lot of power to auto attack heavy champions.

Blade of Ruined King

Blade of Ruined King is must have on Jax it's a great item that synergizes extremely well with your passive. The lifesteal and armor (that your ultimate converts from the Blade's AD) also add to your survivability, together with its active ability.

Randuin's Omen

This item gives you a load of health and armor. The 70 armor gained from this item will give Jax much needed survivability. Also passivly slows the attacker's attack speed by 10% for 1 second if you are hit by a basic attack. It's active slows the movement speed of nearby enemy units by 35% for 2 seconds + 1 second for every 100 armor and magic resistance you have. With this item you should be able to win almost every 1v1 fight.

Spirit Visage

Spirit Visage is a very good item and gives you a lot of MR and HP. I use this item so I can resist AP champs, so I don't get bursted down to quickly. But it's also really good with the it's 20% drain effect.

Trinity Force

This item is great on Jax, every time you use your Empower it does that much extra damage. The slow keeps enemies from escaping, and the attack speed boost, health, and AP all benefit every aspect of Jax.

Trinkets

After you have Wriggle's Lantern you can go for a Sweeping trinket because of the extra ward you get. However, if you feel like you need the vision and mobility more than an aggressive ward removal tool, feel free to keep your warding trinket.

Early Game

In the early game, you want to gank often. As long as you have Q and E, you can gank, having Red Buff helps, but you don't need it. Lanes that have opportunities to gank are lanes that are pushed towards your side, have summoner spells on cooldown, and your laners have sufficient resources (HP, mana, cooldowns) to follow up on your gank. Lanes that need ganks are lanes that are behind either from the lane match up, or assistance or level one fights in the enemies favour. Whenever you do not see an opportunity to gank or have to put a lot of pressure on a certain lane, you should take the opportunity to farm your jungle or go to base and buy if you have money for a big buy. Look at Tactics section for how to gank efficiently.

You can pull off counter jungling with Jax as long as you have a ward to Leap Strike to just in case you get caught (read the Tactics section for counter jungling info). In addition, if one of your laners require you to hold the lane while they back, do so as well. You also want to control both your blue and red buffs, as well as the enemies. Timing the respawn time or knowing the general respawn time of these buffs are crucial for doing so. Control of Dragon also becomes a big factor and whenever your team has a safe opportunity to do so, you should take it.

Mid Game

The mid game is quite similar to the early game. However at this point, you must keep on farming, it is very common to be overwhelmed with trying to gank and end up falling behind. and you have to be in position to gank and contest Red and Blue buffs, Dragon, and Baron. The same things you do to control those, applies in the mid game, time the respawn of the neutral objectives and buy wards to make the job safer. You are still able to gank very well due to your crowd control and high damage, but you do need the help of an ally if you expect to kill a full HP target unless they are really over extended. As Jax however, you need farm versus support type junglers, since you scale a lot better with items than those type of junglers. For that reason however, jungle Jax is not commonly seen as it is difficult to achieve that much farm in the jungle position unless one of your solo lanes roams and you are able to farm their lane and the jungle. If you are able to have a chance to split push a lane, you are in a great position to be a big factor late game. However, keep in mind you won't be stacking your Feral Flare in a lane. The difficulties you face in the mid game make wriggles that much more important.

By speeding up your clear through Wriggles you are less likely to fall behind and depending on how well your team is doing you can farm both the jungle and lanes when they are off ganking/ need someone to push for them. In the mid game if you have achieved your BOTRK you can easily duel/catch out caught enemy squishes. Good use of your ultimate to tank out fights will lead your team to victory. It is always better to use your ultimate right away than to use it late. It has a short cooldown so take advantage of it especially when you have blue buff.

Late Game

In the late game your team wants to group and gain Baron control and secure objectives, whether towers, buffs, or Baron. With proper wards and map control, you are able to do this with ease as well as get catches. Someone on your team, most likely the support, needs to have a Sweeping Trinket to deny the enemy map vision and get map control. In team fights, you are looking to dive the enemy carries due to your tankiness, damage and gap closer. You need to keep farming whenever you can as Jax due to his good scaling. You will scale better than your top lane. Look in the Team Fighting section below for info on team fighting. Jax is very good at soloing kills into the late game. You can go solo and search for lone opponents to prey on. Keep an eye on the map to find your victims. If your team is far enough ahead by now you have have them hold off a 4v5 push while you split. You can most likely dive anyone who comes your way.

Team Fighting

With Jax, you want to be diving their carries in every fight, or focusing down a single target with your team. Using Counter Strike in a team fight is tricky and requires experience to learn. You absolutely need to use it at the right times in order to negate as much burst/damage as possible. In addition, knowing whether to keep it going, or cancel it for an earlier stun is situational. If you are able to get the stun off and kill or drop them so low they have to be extremely cautious and be zoned from the fight, activate the stun and bash them for a chunk of their health pool. Try to spread out your ultimate and Counterstrike in order to negate as much damage as possible, as dodging attacks while having increased armor just wastes your ultimate. Make sure you do not cast your ultimate too late.

Having CDR makes jax a much better team fighter as he is kited less and dodge is up more often. If you are your teams primary tank I suggest you do not cancel your dodge early as you need it to soak up as much damage as possible. Also try to catch as many important targets as possible with the stun. The more you dodge autos the more damage counter strike does up to a cap.

Split Pushing

Jax is an excellent split pusher, as his passive will help him bring down turrets really fast. However, always keep in mind that you are the jungler, which means you shouldn't be split pushing while Dragon or Baron are up. However, while Baron is down you can go bottom lane and push hard if your team can hold mid 4v5.

How To Gank Efficiently

To gank with Jax, walk into the lane and use Counter Strike when you get close to Leap Strike range. This is since Counter Strike has a delay in between the cast and being able to activate the stun. Leap Strike to the target and immediately use Counter Strike for the stun. Attack, and use Empower for an auto attack reset and keep attacking the target while side stepping and use your skills when they are up. Having a team mate who can follow up with a slow will increase your gank success.

However if the enemy has a large numbers of escapes it is better to wait out their escapes then start your counter strike into leap stun. Sometimes if you leap to early the enemy will just use their escapes to get away. By walking up to them and applying red buff then saving your stun leap after their escape you can secure ganks much more effectively.

Jungling Route 1 (Blue side)

This is a quick way to get level 3 with double buff and be healthy enough to gank.

Jungling Route 2 (Purlpe side)

Blue > Wolves > Wraiths > Red > Gank > Base / Jungle depending on how the gank went

Similar to the route on blue side, you wanna get level 3 with double buff and be able to gank mid or top.

Pros

Strong in 1v1 fights and with Bilgewater will win nearly every fight.
Good crowd control.
Beats most top laners.
Tons of damage without much need for offensive items.
Large Item pool.

Cons

Can be locked down in fights.
No Sustain Early.
Vulnerable without ultimate and Dodge.
Relies on Attack speed for damage.
Can get mana hungry if he abuses leap strike early.
Mostly single target damage.