Jax is quite easy to play and when played correctly you can do mass amounts of DPS. He is a good counter for a fed AD carry such as Caitlyn Vayne or even Graves . Finally Jax is just all around fun to play.

Good Times
When ever they are a heavy AD carry team. Such as Ezreal Graves Gangplank Sion so on and so forth.
Also Jax is good against a team of squishies. This is chacters like Fiddlesticks Lux Brand etc.

Bad Times

When ever they are a tanky team with a lot of tanky DPS such as Shyvana Jarvan IV Yorick and especially not Garen all of these characters counter Jax pretty hard especially Garen

Another bad time to pick Jax is when ever they have heavy CC this would include characters like Morgana Sejuani Skarner Riven etc.

Jungle

I find this the best place to play Jax because your jungle is extremly fast and you can gank before level 6. This puts you ahead usually of Warwick Skarner etc. Also your ganks are very strong usually once you get 1 good gank off you can start snowballing.

Solo Top

This is another good spot for Jax because a lot of the AD carry like characters that go top you can beat once you get Grandmaster-At-Arms. These champs include Gangplank Sion Tryndamere etc.

Double Bruiser Bot

Ok you should only go double bruiser bot with a really tanky bruiser such as Alistar Jarvan IV Sion and you do this to counter an AD carry and a Support.

Starting Items in Lane

These are the items to start with if you have an even lane. Teemo would be an example of this.





This is a good item to start with if you are going to be doing most of the harassment. And you think you can sustain along time. Singed at solo top would be an example of this.







These are the items to get whenever you are going to have a really hard lane. Garen Good example of this.

Core Items

SUSTAIN





DPS


or


DEFENCE

End Item

DAMAGE ATTACK SPEED CC





ARMOR AND DAMAGE





Health

WHY GET THESE????

This item is good for countering AD carry because of the armor you get. Good for sustain. I say this item over the Hextech Gunblade ever since it's nerf just hasn't been doing it's job as well.





This is your main DPS item. It stacks perfectly with Jax it gives you hybrid damage. And it gives you attack speed as you attack more just like your Relentless Assault





This is the boot to get when ever you go against an AD heavy team.





This is the boot to get when ever you go against a lot of AP or a really heavy CC team. Really this is usually the boot you are going to get.





This item will give you extra attack damage, attack speed, and a possible CC.





This item is basically built off of the Frozen Mallet to give you extra damage. Also it gives you armor which counters AD carries.( You will see Jax loves countering AD carries)





This is the thing that will keep you alive a lot better. Also it will give you more damage to your atma's implaer.

Defence Masteries

Hardiness Armor to counter AD carries.
Durability and Veteran's Scars It gives you health for sustain and extra damage for your Atma's Impaler
Indomitable To Last longer in battle.
Enlightenment The less time between cool downs the more damage you do.
Juggernaut The more health the more damage to your Atma's Implaer

Offence Masteries

Brute Force The damage
Alacrity More attack speed more damage
Weapon Expertise Armor pen for those tanks

DO'S

Cleanse So you can try and save yourself
Flash To get away
Teleport To teleport back into fights(Off wards or Minions)

DON'TS

Promote You can push well enough on your own.
Surge You don't really need it you can use it but I never would trufully.
Clarity So many better uses then this just don't bother with it.

Main Options

Greater Mark of Attack Damage For jungle helps clear camps faster, for lane last hitting.
Greater Seal of Armor Armor to counter AD carries, and last longer in jungle.
Greater Glyph of Attack Speed More speed for Jax more damage in a shorter times.Fast clear time.
Greater Quintessence of Desolation Armor pen for those Tanky characters.

Other Options

Greater Mark of Desolation For the armor pen.
Greater Seal of Vitality For the health.
Greater Glyph of Scaling Magic Resist Magic Resist
Greater Quintessence of Attack Damage Damage

Jungle Route

Start off with a Vampiric Scepter and attack the Wolves using you Counter Strike







Get a small leash and then you your Counter Strike and then Smite to kill. Now get Empower








Hit the blue wraith then Counter Strike to Empower








Use Counter Strike to Empower then your going to unlock Leap Strike and go back. Buy boots.








Now go get Red Buff Smite to kill it. then gank

Tips

When ever using Counter Strike always hit first then start it.

When Ganking make someone CC first then you gank it will make it a lot easier to hit your Counter Strike

DON'T BABY SIT A LANE. No point you need to be farming if the gank fails don't hang around go farm.

Focusing

As Jax you need to be constantly focusing there main DPS. Also you need to know who to hit your Counte Strike on. Good options for this are Mordekaiser Fiddlesticks Sion Casseopia .

Cleanse

Using your Cleanse is a major part of team fighting. Time it whenever there main DPS is trying to run from you or there initiator targets onto you.

Initators

Don't ever initiate a fight EVER. That is an important part of Team Composition for Jax you have to have a really strong initiator such as Alistar Blitzcrank Nautilus etc.

Grandmaster's Might

You should use this when ever there team starts attacking you. Don't use it whenever you jump into battle because by that time they are usually still focusing someone else.

Other Options

This is good for whenever you are just getting destroyed by and AP nuke. Such as LeBlanc Katarina





This item will help you out a lot with straight CC and sustain. Really only ever get this whenever your team is a little on the light side for CC.





This isn't my recomended item but other Jax players I talk to say AP but I usually do better then them so I just disreguard them but if you want to go AP go for it. It really helps out your Grandmaster-At-Arms and Empower .





This is for whenever you are basically back packing your team through the game and if you die then you'll lose. Also if they have a Karthus that keeps using Requium on you or any Universal Ultimate Champion.






This is for AP heavy Jax it will help deal a lot of damage with your Empower and Grandmaster-At-Arms .





This is good because when ever you use a spell it will slow them because of it's passive and it gives you magical DPS.





For a really heavy AP team. This shouldn't replace a Frozen Mallet or Warmog's Armor it should be replaced for a Lich Bane or Trinity Force





This is good to counter Mundo for it's passive.





This is sort of a replacment for your Wriggle's Lantern but I wouldn't suggest it because you can Leap Strike to it's passive Sight Ward

PROS
You are the Greatest 1v1 Champion in the game. ( Fiora ain't got nothing on Jax )
Jax is easy to play
Jax does MASSIVE damage
Jax is so sexy ;)


CONS
Tryndamere is still scary
CC wrecks Jax

Update 5/3/12
Changing the Lich Bane for a Trinity Force . Looked into it I misunderstood the passive. Changing the Frozen Mallet for a Warmog's Amor because you already have a chance for the passive with Trinity Force . Also Name change from "Jax The Grandmaster at Arms" to Jax World's Sexiest Bruiser"

I must say this was an experience writing this. Any tips constructive critisism would be nice and finally bye.(BTW didn't draw this picture)