I've been playing LoL for 6 months now and my main champion is Jax.
With this build you can go both lane and jungle.
Also I wanted to keep this guide simple and not too long.

In LoL world hybrid is the champion that needs both attack damage and ability power to be strong, or to make his abilities stronger.
People are playing hybrid style because it is hard to counter.
If you go AP you will easily be countered with items like Force of Nature or Banshee's Veil. Also if you go AD you will easily be countered simply with Thornmail!

Hybrid items in the game are only Hextech Gunblade and Guinsoo's Rageblade but you can also combine some AD and AP items to build your hybrid champion successfully.

OLD Relentless Assault
Gains 3 health per point of Attack Damage received from items.
Gains 2 health per point of Ability Power received from items.

NEW Relentless Assault
Every time Jax performs an auto-attack, he gains 4 / 6 / 8 / 10 / 12 / 14 % attack speed. This buff lasts for 2.5 seconds and stacks up to 6 times.

OLD Counter Strike
Passive: Increases Jax's dodge chance by 10/12/14/16/18%.
Active: Jax stuns surrounding enemies for 1 second and deals 80/100/120/140/160 (+0.6) magic damage to them.
Usable only within 7 seconds of dodging an attack.

NEW Counter Strike
Jax enters a defensive stance for up to 2 seconds, dodging all incoming basic attacks and taking 25% less damage from area of effect abilities. At the end of the duration or if Counter Strike is activated again, Jax deals physical damage and stuns nearby enemies for 1 second. This spell deals 20% additional damage for each attack dodged to a maximum of 100% increased damage.

OLD Grandmaster-At-Arms
(Active): Jax gains ability power equal to 25/45/65 (+20% of his ability power) and attack damage equal to 25/45/65 (+20% of his bonus attack damage) for 6 seconds.

NEW Grandmaster-At-Arms
(Active): Jax gains armor and magic resist.
Armor Bonus: 25 / 35 / 45 (+0.3 per bonus attack damage)
Magic Resist Bonus: 25 / 35 / 45 (+0.2 per ability power)

Greater Mark of Alacrity
These runes fit in great with the passive, especially in the jungle.
1.7% x 9 = 15% AS

Greater Seal of Vitality Greater Seal of Vitality
This gives a nice amount of HP combined with Durability
(19.44 x 9 = 175) + 108 = 283 HP (at level 18)

Greater Seal of Resilience
You must teke these if you want to survive the fight with red golem.
1.41 x 9 = 13 Armor

Greater Glyph of Force
Gives a lot AP which is important.
3 x 9 = 27 AP (at level 18)

Greater Quintessence of Fortitude
Really helps on start of the game, also when you combine it with Veteran's Scars you will have bonus approx 100hp.
26 x 3 = 78 HP

Greater Quintessence of Vampirism
Helping in jungle, it's more than a half of Vampiric Scepter.
2% x 3 = 6% Lifesteal

Is a must:

Flash
This spell is important for every champion including Jax. It gives you ability to run away easy flashing over a wall.
If you're not Level 12 (don't have Flash), take Ghost.

Very good:

Smite
If you're going jungle take it.


Ignite
It will make you win a lot of early fights, especially against champion that have inbuilt healing effects.


Teleport
It's really good to defend turrets or just to get in lane faster when you're solo top.


Surge
This spell is too strong on hybrids including Jax.
Use it combined with your ultimate when you're destroying turrets or going into fight with someone. Also don't go in 1v2 until you finish your hextech.
It gives you 10-78ap and 35% attack speed for 12 seconds!


Cleanse
When enemy have champions such as Amumu, Morgana, Nasus, Leona or when you just want to get rid of Ignite or Exhaust.


Exhaust
Not too important on Jax since we have Rylai's Crystal Scepter but it could be handy when you combine those two.


Heal
I don't find it important but good if you want to heal bait.

Never:

Clarity
You are not that mana hungry and you have mana per level in masteries.


Revive
We do not die!


Promote
Use this when you want to have big minions. Super effective! (lies)


Clairvoyance
If you want to go support with this guy take it! (never go support with him)

Jax is a great champion because of gap closer that he has - a.k.a. this skill. We will not use it for poking since it's costs a nice amount of mana, for that we will use Empower. It's much more mana cheaper and the damage is higher.

Low cooldown and high damage, also works great with Sheen!
Take this at level 2 at level it up as much as you can, since the cooldown is reduced and damage is increased on level up. You can also try turning it on and then jumping on enemy with Leap Strike for bigger burst damage.

When both laning and jungling take it as first skill. Use this skill to stun and enemy when ganking or to dodge and deal some damage to buffs or enemy. We level up this skill one on the start and leave it till the late game, because we don't use this skill to deal damage.

Level up your ultimate whenever you can and use it in fights or when you want to escape. It gives you nice amount of armor and magic resistance.
The bonus armor is based on your AD and magic resist is based on your AP!

STARTING ITEMS:

Boots of Speed
Main and the best starting for most of the champions.
This is a great starting item if you're playing against champions that uses skillshots to deal some damage to you, like Kennen or Lux.
Also good against champions that can't deal damage to you if they are not right next to you like Garen and Nasus. Also if you get this item buy 3 Health Potions!


Vampiric Scepter
Tested, it gives you the best result when jungling with Jax.
Although you need to have jungle runes.


Cloth Armor
It gives you more sustain in lane and great fro rushing Wriggle's Lantern.
Also you get 5 Health Potions on start if you pick this item.


Regrowth Pendant
Gives you great amount of sustain in lane and you can also but 1 Health Potion.
This item you can only build in Philosopher's Stone which is great item since you will get both HP and mana regen. plus additional 5 gold every 10 seconds!
Also it's not bad to buy one more gold item like Heart of Gold. When you lack slots for other items just sell these too.

CORE ITEMS:

Wriggle's Lantern
The item gives some life steal, attack damage and nice passive for farming (20% chance to deal 425 magic damage to monster/minion on hit). The active gives you free ward and that is more than useful for solo top! Also main item if you want to go for jungle.


Berserker's Greaves
If you are doing okay these boots will do the best.
Attack speed is amazing on Jax because of his passive on ultimate, every third attack he deals extra magic damage.


Trinity Force
Procurance - deals extra damage after using and ability (150% of base AD) and slow because of Phage works amazing with Empower.
Also that extra health, mana, move and attack speed is a great bonus.


Hextech Gunblade
This is a great item for hybrid champion!
It gives you every single offensive thing: Spell vamp, Life steal, AD, AP and Active (dealing 300 magic damage and slows the target champion by 50% for 3sec, 700 range and 60sec CD). This item is giving you a lot of sustain and you will be able to go 1v1 with almost anyone!


Guinsoo's Rageblade
This is an amazing item for Jax, gives you AD and AP but the thing that makes this item so awesome is it's passive. You gain amount of AS and AP per hit.
That means that you will reach your every third attack faster and it will even be stronger.


Wit's End
Like I already said attack speed works great with Jax, and with bonus 40 magical damage every hit and extra magic resistance this item is amazing!


Rylai's Crystal Scepter
This is really good item because it gives you health and some CC early on.
Also we take it for some damage.

FINISHING ITEMS:

Guardian Angel
As finishing item if you are focused in team fights.
Gives you tons of armor and some magic resistance, and great passive!


Banshee's Veil
Great counter to heavy AP carries on enemy team, spell shield is a great thing but it can be broken easy.


Madred's Bloodrazor
Great pick if enemy team is really tanky.
But also this item is really expensive, I wouldn't recommend it.


Quicksilver Sash
Buy this item when you have Mordekaiser, Vladimir, Nasus, Malzahar... in enemy team to counter their abilities. Activate item when they throw some of their abilities on ya.

SITUATIONAL ITEMS:

Ninja Tabi
If solo top champion is harassing you bad with AD and he has no CC, buy these.
Instead of Berserker's Greaves.


Mercury's Treads
Against heavy CC teams and especially champions like Rumble.


Warmog's Armor
If you want more tanky and AD Jax buy this and Atma's Impaler on the end. Also when you get Warmog's Armor farm as much as you can, because the passive gives you HP and HP regen. per minion kill!


Atma's Impaler
On the end sell Wriggle's Lantern and get this for more damage and some armor/critical chance. Also you get AD from 1.5% of your maximum health!

Doran's Blade
+10 attack damage +80 health +3% life steal
Gives some AD and health, also if you stack it you will get some life steal.
If you choose to start with this item, when you're out of HP go back and buy Health Potions. Also if you pick Doran's items for start be sure to have Teleport as summoner spell.


Doran's Ring
+15 ability power +80 health +5 mana regeneration
You really don't need AP on the start of the game, and you really don't have mana problems.
It gives you some health but I wouldn't recommend this item. It was viable before but not anymore since Jax got new passive.


Doran's Shield
+10 armor +120 health +8 health regeneration
This small amount of HP regen. will give you a little bit of sustain but it's not worth buying or stacking it. It's much better to start either with Regrowth Pendant or Cloth Armor.

Early game:
- You can start with Cloth Armor, Boots of Spped] or [[Vampiric Scepter. If you have enough money for potions, buy them.
- Poke an enemy with W and jump back with Q if he thinks of attacking you. When you scare him a little bit just keep farming and be aware of ganks, use your Wriggle's Lantern to ward river brush whenever possible!
- Let the enemy push your lane if your jungle wants to gank and possibly kill the enemy.

Mid game:
- Finish the top turret and gank/push other lanes.
- Help your team get dragons.
- Search for enemies that are pushing alone since Jax is great in 1v1 or 2v2 fights. If enemy decides to fight activate your Grandmaster-At-Arms and Surge, if you have it.

Late game:
- Focus AD/AP carries in team fights.
- Help your team getting baron and push lanes some more, especially if you have Lich Bane. When your team gets baron just try not to waste it.
- If their team starts going offensive be sure to protect your turrets with someone beside you, never protect turret in 1vs4/5!

Usually in the fights you are not the one who is jumping in first, you should be the second.
Wait for tanks to initiate in the fights, and than you jump in. When I say tanks I mean Olaf, Amumu, Rammus, Shen, Renekton, Nasus...
If the fight is not started do not poke enemies with your Leap Strike, you will just get stunned/slowed and you will probably die.
Also in fights focus only AD and AP carries.
AD Carry: Vayne, Graves, Caitlyn, Ashe, Sivir, Miss Fortune...
AP Carry: Morgana, Malzahar, Kassadin, Veigar, Vladimir, Annie, Kennen...
You are focusing these champions because they are damage dealers in the enemy team, and they are usually very squishy. But don't chase them through the whole enemy team like a mad man.

The fight starts when the enemy team attack your turret/inhibitor or someone from your team.
If your team doesn't want to start a fight just try to stun the one who is chasing your teammate and run away.

- On the map search for enemies that are pushing alone because Jax is doing better 1v1 or 2v2.

- When you buy Wriggle's Lantern you will have free ward and that means it gives you free icon=Flash since you can jump on wards with your Leap Strike! Put ward over a wall and jump on it while they are chasing you, very effective! You just need to practice setting ward and then jumping on it, it's not to hard when you get used to it.

- You can also buy Oracle's Elixir to see and jump on enemy wards while you're running.

- When you want to gank when you reach the brush activate your W, E and than jump on enemy with Leap Strike! That way you will deal a lot of damage and victim shall be stunned.

- When you want to steal baron buff while enemy is taking it just stand behind baron and put 1 ward near baron. Put it while they are attacking baron so they don't kill your ward. When baron is left with 500hp flash in, smite it and jump to the the ward you have placed over the wall.