Why i choose jax

I play jax in practically all my normal games (unless i am testing another champion) and some ranked games depending on the circumstances. The reason for me playing jax is because i love the damage output, the tanky side of him, and the pro E dodge and stun. The Dodge and stun spell is the main reason i love jax, Even after the nerf this spell continues to dominate other AD champions simply because, if a full AD champion attacks you (like tyrandamere) then this spell acts as a complete immunity for over 3 seconds (including the stun) and yet you still continue to drain their health away. jax is also good against AP champions as such due to the stun, Excellent survival ultimate, and good escape material, and so you will always have the upper edge in movement.

Pros

Excellent tanking survivability
Great damage output early mid and late game
Excellent stun throughout the whole game
great escape and chase jump

Cons

No high damage ultimate
Rageblade takes time to get to 8 stacks

Start with boots of speed and 4 health potion as the damage doesn't need to be improved really at this level, and requires speed as the jump has long CD low level.
Try not to attack enemies until level 2 when you have jump, as jump not only helps on getting them, its also useful if they run away, and so if you were to get them to '12' HP at level 1, well sucks to be you.
sustain the lane by getting last hits on minions, in games i don't really mind letting the jungler gank other lanes more than mine, as i tend to do fine without and stun + jump away if i get ganked, so you can actually push the lane considerably if you want, to make sure you have time to recall whenever, and get back to the turret before the enemy reaches it and starts taking HP off the turret.
Jax is also not a bad 'turret diver' When you get to around level 7, due to the tanky structure he will have at that level, and the superb defensive ultimate, so if an enemy had around 300-500 health at this stage and you were near full, it would not be a bad idea to jump and stun them, as they will probably die.

Now hopefully at this stage (level 12 or so, people are coming out of lanes regularly) You will have build up 2 or 3 kills depending on the situation, if this is so then you will hopefully have trinity force or at least be getting there, So you will be pretty good at dealing the damage and have a high movement speed, If so then you can move around out of the lane a lot, try to grab a red buff if the jungler doesn't mind as it will help kill the enemies who run away, help the other lanes if they are getting a bit pushed, and help kill the dragon if your team helps, at this stage you can almost act as a second jungler for mid, as the jump and stun technique will almost always kill an enemy, even at full health, if your mid helps you do so. however try not to focus too much on being this 'second jungler' i have introduced, as the enemy team will soon realise what you are doing and will soon make it their most effort to kill you as many times as possible.

At this stage, hopefully you will have racked up a good 7-8 kills, and will surely be dominating most people, if not then go farm the jungle a bit and act again as this 'second jungler' and it will surely get you some kills. Jax is excellent for teamfights especially if you dont have a tank (you are the tank) then you can start the fight and hopefully the enemy team will attack you, Either way, Activate the E ability and start kicking *** by spamming W whenever its available. This way, with a good team, you are sure to win teamfights with a good ADC and AP carry. As the damage will be amazing combined with yours. You can also try backdoor a lot (for those of you who don't know what this means, its basically going top or bottom lane and getting a turret down while the enemy team is focused mid) Jax is a good backdoor champ as of the terrific movement speed at this stage and good escape mechanism if the enemy team happens to catch wind of what you are doing, If you are losing at this stage and the enemy team is in your base, do not hesitate to try push them out slightly by stunning them and such. however do not try and jump the whole enemy team as they will instantly target and kill you.

Boots

I have started with boots of speed as explained in the early game section as a boost to my movement early game and so the enemy team cannot constantly poke you.

Trinity force

I have chosen trinity force as my first big item due to the great damage output it provides, the movement speed bonuses, and the excellent boost to your W spell (deals much more damage)

Guinsoo's Rageblade

Rageblade is next as i like the attack speed build it gives as it stacks very well with your passive, and also gives an increased boost of ability power which will help lots late game. Also if an enemy happens to get you below 50% the passive will kick in, giving you Great advantages such as 20% attack speed, this extra will help lots and if the enemy is not fed, you should win the fight.

Hextech Gunblade

Gunblade is next as it gives an increased boost to attack damage and ability power, Jax is addicted to both of these. Gunblade also gives 20% spellvamp and 12% lifesteal, These are very good in maintaining jax's health, and if you happen to get ganked but get away, you can easily go into the jungle attack a few beasts, and restore some of your health.

The Black Cleaver

Black cleaver is next as i think the armor penentration helps massivly, It will build so fast due to jax's great attack speed and once at 25%, even enemies building in thornmail or such, will suffer just as much as a squishy non armor champion. Black cleaver also gives 200 health, Very valuable to jax as at this stage, you will have more health than most damage champions in game, and still have loads of damage.

The Blade Of The Ruined King

Lastly the blade of the ruined king, Most guides chose a different item to this such as bloodthirster or atma's impailed, However i think the blade of the ruined king benifits jax the most as it gives a good boost to attack speed and damage, and also deals 5% of their health per second, helping your damage greatly when W is on cooldown, The item also gives 15% lifesteal, i think this is great at this stage as you will have Massive attack speed and damage, and so this will certainly sustain your health against most champions.

Situational Items

Warmog's Armour

This is another good choice for jax as it will make you extremely tanky meaning you will be able to jump in to large crowds of enemies without worrying too much, I have not included this in my main build as i think that having trinity force and the black cleaver gives you enough health already, In games i can solo baron as soon as i hit the black cleaver on my usual setup, and also do not have to worry about getting ganked by the whole team lategame.

Atma's Impaler

This item should be taken if you are taking Warmog's Armour as it will give a great increase of attack damage, However i do tend to not use this item as it gives no advantage to my hybrid guide and i feel the armor is not generally needed.

Maw of Malmortius

Maw of Malmortius is a great item to buy when playing jax. Since normally your opponent's AP carries will have a high damage output. This item will give you some magic resistance and a 400 magic damage shield. Which is really nice against AP carries.
This is also an offensive item with 55 AD and the passive which gives you 1 AD for every 2.5% of health you're missing (up to 40 AD). With Jax's ultimate, this item will give you some extra armor.
It will give you both magic resistance and armor in a total of 42.5 to about 65 defensive resistance with jax's ultimate. This item is pretty good for your sustain, durability and it comes with some offensive stats too.

For masteries I choose to go 21-9-0. Since jax is a great damage dealer you need the offensive masteries. Jax is also quite tanky, so i take the 9 points in defensive for the 3% health mostly, but also the reduce of all damage by 2 i think is great.
Spell weaving and blade weaving are great masteries and if you want to change some points around, i suggest keeping these 2, its like they are built for jax, increasing all his damage output.

Mark Of Armor Penetration

I choose this as my Mark as i think it gives the most damage throughout the whole game, in the past i have tried mark of attack damage and mark of attack speed, and altogether i have done much better with armor penetration, it helps with doing tons of damage, and also helps against enemies that build armor, This item is also excellently combined with The Black Cleaver (see items) to do outstanding damage to any champion, regardless of the armor they build.

Seal Of Armor

I choose this as my seal as it gives a boost to survivability throughout the game, This greatly increases the chances of you obtaining first blood as this guide is designed for solo top players, therefore you are most likely to go against an AD champion or an AP who regularly auto-attacks therefore these Seals make sure you take minimum damage.

Glyph of scaling magic resistance

Glyph of Scaling magic resistance - magic resistance pr/lvl - This will help you against AP champ's deadly bursts at mid to late game.

Quintessence of armor penetration (1)

I Choose this as 3 of my Quintessences as the same reason of my marks, to give a good increase in defeating enemies early to late game, and also helping against those enemies who build armor.

Quintessence of Attack Speed (2)

These runes are situational if you do not like the armor penetration runes. I choose these 2 as my Quintessences as i think they give a good boost in attack speed which is vital to jax, and also these quintessences combine really well with Jax's passive at low level, giving you more attack speed than most other champions, really giving you the upper hand in obtaining the first blood.