With the recent buff to hybrid items, the rework of jungle items which favor aa reliant champs and new masteries, jax is really strong right now.

I used to be a yi main in season 5, however, due to the Blade of the Ruined King's nerf, yi is not having a very good time in the jungle. While searching for a solution for the dramatically decreased winrate on my yi, i, happen by chance, found the cowsep's ap yi build, with rageblade and gunblade. However, statistically, the ap yi would not be an optimal use of stats, you need a an hybrid champion for that. Then, i asked myself, why not just using that build on Jax?

Jax has always been known as a premier late game duelist, however, his struggling early laning phase makes people think twice before picking him top. If played as jungler however, that weak laning phase is avoided and with recent items changes that give give hybrid champions early and massive power spikes, Jax doesn't really have a weak phase in the game right now.

Jax's scalling is insane, if you're tired of ganking non-stop, only to find the ganks fail due to teammate's mistakes or losing games because of your teammates throwing leads ignoring your calls and request, or if you're fed up with the complains that your teammates keep throwing at you when they're the one that keep overextending without wards. They'll then ask "why are you not ganking? X has been ganking me 5 times, stop farming, you scale like a potato." - said the 0/7 darius top.
Then jax is for you. Jax can single-handedly carry the game by himself, wrecking the enemy's backline within seconds, winning the teamfight before your team even notices what happened.
In the end, only result matters, everything else is irrelevant.

Greater Mark of Attack Speed & Greater Quintessence of Attack Speed this the must have runes on any auto-attack based champions, which can also be used on tryn, yi. The reason why these runes are so good right now is because of the new Hunter's Machete doing 20 on-hit bonus dmg, which make your clear a lot faster.

For the mastery, you can spent points on Resolve tree instead of Cunning if you want to be a little tankier and/or like to have 15% reduced cd on summoner's spells( can be paired with cdr boot)
Currently, the Precision mastery and thunderlord is quite OP, therefore, before the upcoming 5.24, where fervor gets buffed and precision get nerfed, Cunning tree is the way to go.

Guinsoo's Rageblade
Really cheap and powerful, serve as an early game power spike, with this item built, you're guaranteed to win 1v1 and 2v2.
Hextech Gunblade
For Jax alone, this item isn't much better than the old gunblade, however since the Blade of the Ruined King's nerf, this is a reasonable purchase. Moreover, this item is quite cost effective and the active is very useful as well.
Trinity Force
Classic core item on jax. However, with recent changes, this item is not necessarily a must buy , still an efficient purchase on Jax and is a decent mid-game power spike.
Maw of Malmortius
I absolutely love the Lifeline passive on this item, attack speed +lifesteal + spellvamp, until out of the freaking combat, meaning that this passive can last for an entire teamfight, and a magic dmg shield, which scale with bonus MR (stack with your ult) on top of that, you can't possible ask for more than that, this item is a must have if you're playing against a strong ap.
Nashor's Tooth - suggested by Riley Reid
Really nice item, provide AS, CDR, and on-hit magic dmg, which synergizes well with devourer and Guinsoo's Rageblade. This item, however, with AS runes, devourer, rageblade and your passive with give you an excessive amount of AS ( capped at 2.5) and building glass cannon may not always be the best idea, so, build it with care.

The CDR build: - Special thank to zekirora
this is an extremely useful build if you're going to need your counter strike really often( your team lacks cc or initiation, playing against multiple aa reliant champs etc). At 40% cdr, your counter strike is at 4.8s cd or a 2s dodging and 1s of AOE stunt, meaning that the like of tryn, yi, adc will only have 1.8s between each counter strike for any sort of counter play. The slow from Iceborn Gauntlet and Hextech Gunblade is to guarantee that the target will not be able to kite. This build can only be countered by casters, especially single target assassins, you may want to focus them first in teamfight if they're relatively fed, since the rest will not be able to do anything to you with your counter strike except for them.

Warmog's Armor + Sterak's Gage - Special thank to zekirora
This combo gives you a lot of raw health to work with, along with the bonus resist from ult, you'll be impossible to take down for flat true dmg dealer like yi, irelia, darius, chogath.

Leap Strike Really versatile ability with low cd and decent base dmg, you can use it to gap close, escape by jumping to a ward or your teammates as well as maneuvering in teamfights, you can start a fight by flanking from side with your counter strike, then jump back to your teammates to avoid and dmg when Counter Strike is on cd, and look for opportunity to jump on important targets.
In late game, conserve this ability as a gap-close, not an ability to do dmg, if you can save your Q, do so.

Empower auto attack (aa) modifier, resets your aa timer. Basically, you do a normal aa, then W for 2 consecutive hits

Counter Strike Dodge all aa and on-hit effects, like gangplank's Q, ezreal's Q, and recently Nasus' . Still not yasuo's Q though, even though his Q is counted as an aa, hell, not even teemo's blind can't stop that basterd's Q from doing dmg, i suspect that Riot forgets that on purpose. After 2 second of dodging all the **** thrown at you, you'll stun the enemy for an extra second, meaning that you have 3s of no counterplay if your enemy only do dmg by aa.

Grandmaster-At-Arms
More dps + more resistant. Fairly low cd, which should be exploited as much as possible.
8s duration is generally enough to last an entire fight, therefore, you generally should active your ult as soon as you're about to take take dmg, however, if you're playing against aa-based champs, you can active ult after Counter Strike to make the most use of it.

I.After first-back.
-This is the point in the game when first blood usually occurs which will snowball a lane out of control, look for a lane with enemy going ignite, your lane is pushed, and the champs in the lane has the tendency to fight early, like riven, renekton, irelia, olaf,..v..v.., you'll have to rush there ( skipping jungle camps) if the laners are really low, a kill can quickly swing the lane into your team's favor.
-Instead of going back after blue, you can go for a full clear to get stalker's blade at first back, doing this will get you the stalker's blade earlier for easier gank. However, unless you have a good leash or start with 3 health pots, you'll be low on health and vulnerable to invadewhen you reach top side jungle and likely won't be able to participate in gank/counter gank early. So, do this when you're not playing against aggressive jungler and/or having a farm match-up top lane.
Special thank to Orange Sparks

II.Pre 6
-After the first blood, if your lane is behind, ask them to play safe, sacrifice a few minions, wait for the lane to be pushed to your side, gank for them only if convenient and it is safe to do so( the enemy's jungler is spotted elsewhere on the map), remember that, when your laner is behind, if the enemy's jungler show up for a counter gank, you are very likely to lose 2v2 pre 6, since jax is pretty weak up to that point.

III. Post 6
-You hit a decent power spike, and can do a lot more by then, therefore, this is the time that you can get a lot more kills, especically after getting stalker's blade for the extra CC, gank the lanes that are overextended, lack escapes, squishy, or your laners have good set up for ganks, after 6, several champions hit huge power spike as well, you may also gank those lanes.
-If no ganks is really convenient or too risky since you and/or your lanes are behind, keep farming til sated, look for lanes to cover to get the extra gold and exp, jax scales really well with items and level, so, keep playing.
-Try to take the dragon whenever possible, 5 stacks (2 more if scuttle crabs is also taken), it will speed up your sated.
-If a lane of yours is snowballed out of control in favour of the enemy, you better leave that lane alone, perhaps try to ward the river nearby to prevent the enemy's jungler from diving and avoid ganking that lane, If you try to gank that fed lane early, what will happen is either you'll trade 1 for 1, with either you or your laner dying, neither of them is good. If you die, you'll fall behind, leaving other lanes under pressure from the enemy's jungler in the meantime. If you laner dies and you got the kill, the laning that you gank for will fall even further behind. The worse case is that enemy jungler or mid laner show up for a counter gank, which promises disaster. Replying on your laners to do well, to outplay his opponent is rather a risky bet to make, if he is actually capable of playing well, he wouldn't have fallen that far behind in the first place.
With enough farm, comeback when you're confident that you can at least 1v1 that fed laner. Only then, you'll not have to rely on your laners to do well do pull off a gank.
Also, if any of your lane is overextending due to them having a significant lead over their opponents you want to farm the enemy's jungler on that side as well, to starve the enemy's jungler as well as provide help and protection for your fed lanes, have wards with you, wards the enemy's jungle as well, the enemy will want to gank the overextending lanes whenever possible.

IV.After sated
This is your next power spike, usually, by now, you should already have Guinsoo's Rageblade, this is the time that you'll want to split-pushing to get other core items and level, ask for a lane to split-push whenever possible.
When split pushing, remember to have a ward to jump on to escape, also, have a Vision Ward in the enemy's jungle as well.
When the enemy team is trying to pressure a lane of yours, quickly calculate teamfight power of each side, if the enemy team has something like amumu, malphite, alistar, yasuo, misfortune, you may want to split pushing as much as possible, without dying, of course.
Remember to keep in mind the time for objectives, you'll want to recall and head to the dragon at least a minute before it spawns.

Ganking with jax is pretty simple, flank the enemy's back with counter strike active, stun them, Q if they flash, otherwise, keep smashing them with.
If all of the lanes are pushed, Look for lanes with strong CC or high burst, Jax's relative high burst can take a squishy target down really quickly.
Spend the rest of your time power farm in the jungle and ocassionally warding the river whenever your warding trinket is up.
Buy a Vision Ward, place in the bush near either of your Red or Blue buff.

Invade:
Start doing this when you have a lead on the enemy team's jungle and can easily kill or make them burn their ult to escape, remember to have a ward in reserve to jump to if things turn sour.

With Guinsoo's Rageblade, you can push wave and take down tower really quickly, you have the abilities to escape ganks easily as well, split when the enemy has a significantly better teamfight than yours or to buy time for your carries to scale up.
Always have a ward in a the inventory to escape.

So, jax is sort of an assassin, who can burst down a target really quickly. Due to your single target dmg nature, it's usually best to focus on taking down the most fed target first. However, if a squishy target is out of position, you can always jump in to take that target out immediately.
You always want as much vision as possible before a fight happen, any mispositioning is an opportunity.
If your team is losing teamfights, find the reason and try to avoid teamfights as much as possible( split pushing), ask your teammates for the reason, if your carries are getting destroyed too quickly, keep the target that are doing that damage, if the enemy is having too much AOE CC, try to bait for the CC to be spent and flank important targets or try to make picks, avoid 5v5 teamfights.

Tryndamere
Only active your Counter Strike when tryndamere is about to hit you, otherwise, you'll lose the fight if tryn has his ult up. Go for the cdr built in this match-up.
In teamfight, chunk tryn down quickly to not give him time to do dmg, apply your Hextech Gunblade's active on him when he uses his ult and focus other target.
If you want to gank a lane Tryn, try to do it at least once pre 6, he is really aggressive and tend to push hard early due to his high dmg nature. After 6, gank him whenever his Undying Rage is down, ask your teammate to time it, it's on 110s cd at 6 and 100s at 11, trynd usually doesn't build cdr.

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Master Yi
Similar to tryn, Counter Strike when yi is about to hit you, also, walk up to him, save your Q so that you can jump onto him when he run away with his ult, trying to dodge your Counter Strike.
If Master Yi is really good, he can save his ult, level his Meditate to lv 2 early, active it when you Counter Strike, and Alpha Strike to dodge your Counter Strike, after that, it's his turn to ult and it will be a dps race afterward. If you're as relatively equal or only slightly behind Master Yi on items and level, you'll most likely win, with your passive and rageblade fully stacked . The only problem is that it will be a lot lot harder for you to chase him at this point, if he decided to run away, since your Counter Strike is on cd and he cannot be slowed with his ult on. Moreover, if he is far ahead of you, that will also be the sequence of couterplay that will win him fight against you, when your Counter Strike is on cd, it will be really hard for you to run away from him or fight him, since Chilling Smite is also weaker than Challenging Smite in most duel to the death, my suggestion at this point is to avoid fighting him til you get a few more items, 4 items jax usually win against 5 items yi.
If you know that you his ult is on cd, you can simply Q directly on him, apply Chilling Smite, to force him to Flash, if he doesn't have flash as well, he is most likely to going to die.
Get Trinity Force and Warmog's Armor in this match-up ( after rageblade, of course).
You can quickly take him out of the teamfight unlike tryn, do that.

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Xin Zhao
Similar to the Master Yi and Tryndamere, walk up to Xin Zhao, wait for him to hit you, then Counter Strike, Leap Strike onto him if he ults or flashes away, if he used both ult and flash to escape and you're sure that it's safe to continue chasing him, then flash as well to follow him, Xin Zhao has no escape after ults and flash.

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Lee Sin
He will look for early game invade, ward the river bush or blue/red buff bush before doing the buff.
He will also look for a lot of early ganks, try warding for your teammates as often as possible.
His scaling is rather poor, if your team don't fall too far behind at the point of which you got Guinsoo's Rageblade, ( and of course devourer), you should confidently win most 2v2, 3v3 against him.

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Rengar
Rengar will look to farm til 6, he is rather vulnerable during this period, you can counter jungle/ invade if your lanes are pushed.
If you're relatively fed, you can start invading his jungle fairly often, rengar doesn't have much escape, and if he burns his ult to run away, he will also not be able to gank for the whole duration of the ult as well.
Rengar relies heavily on his Thrill of the Hunt to gank, which is 150s cd at 6 (120s with 20% cdr), 110s cd at 11 (66s at 40% cdr), Rengar will generally build cdr (usually 20% at 6 with runes and Enchantment: Warrior, and 40% at 11), so keep that in mind to predict his action.
If rengar is fed, prepare a Vision Ward, place in down next to your carries before the teamfight is about to start, it will help to spot and kill rengar quickly.

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Kindred
Early, When her W is active ( there is a big circle around her, which reduces her dash to 2s) avoid fighting near that circle, she can kite you really easily and out dmg you as well. Lure her out of the circle and fight back when it is on cd.
Her ult will prevent anyone from dying in it as well as heal anyone as well, meaning that you and your teammates will not die there during the ult, therefore, save your Counter Strike and Leap Strike while try to do most dmg to multiple targets, with your burst, you should be able to instantly kill a low health target nearby and run away to safety waiting for a chance to re-engage and your abilities to come off cd.

1. Don't flame your teammates, keep talking down your teammates will never help them to do better.
2. Don't give up, Jax is really strong late game champion and and take down turrets extremely quickly, if you ,happen by chance, to catch a carry or so, you can always force several objectives off that, the game is only lost when either your nexus is destroyed, or your team gives up.
I do believe that the above 2 reasons are the top reason why people stuck in low elo, i myself do have a hard time convincing my teammates to keep playing and stop criticizing their teammates . It makes me feel sick and disgust to see people type in "gg" after someone makes a mistakes, i don't believe that flaming, raging, afking, **** talking are what "good game" was supposed to mean.

3. Try to learn from every lost: Your game can be record by requesting a record from op.gg or from a third party program, try to make use of them, occasionally revewing your game is one of the best way to improve