This is my first guide, and I don't consider myself a pro by any measure. I play for fun, and its fun to play Jax. I made this guide more as a way to get feedback from the community, I don't care if I have the best or worst rated build on this site. I almost always SOLO QUEUE, which most likely makes a difference in how I play, I'm sure if i was on organized teams I would play/build differently. I try to build in solo queue to do enough damage people run away, and if they don't run i want enough HP that i outlast them and next time they see me they run anyway. I like winning, but in Solo queue its just not always going to happen, no matter how hard you try. I would like to hear what people think about how i build my Jax, if you use it and like it, great! If not, whatever, LOL.

Pros

Jax ROCKS!!!
He can be played Solo Queue, and usually is
Can take almost anyone 1v1
Can defend towers when needed
Can capture unguarded or take guarded towers

Cons

Focused Hard and often
dominion doesn't usually last long enough to finish builds

Basic Rune set up for Jax as far as I'm concerned. I sometimes go with all Greater Mark of Attack Speed instead of Greater Mark of Desolation Not sure which i like better yet. Greater Seal of Evasion are pretty standard for any jax. Dodge is important. Greater Glyph of Attack Speed are the best i could find for jax, maybe Greater Glyph of Magic Resist would be better, but i dont have those runes yet :) For Quints i use Greater Quintessence of Attack Speed but i like Greater Quintessence of Desolation also. Maybe even Greater Quintessence of Movement Speed movement speed is important, but don't know if i would change it for either of the other 2 quints.

Again, Basic 21/0/9 setup. Good Hands Rocks!!! I use Ghost, so..... Haste I like more damage more than more defense, and going 9 points into Defense isn't worth it to me, just for 2% dodge (you should easily be over 30% without it) and the movement speed after you dodge is nice, but not really needed. If you are getting focused the small amount of Armor or Magic Resist isn't going to make enough of a difference to take them.

I take Ghost and Ignite

Ghost because movement around the map is more important to me than Flash. Flash is good for escaping and chasing, but it's a one time use, and if you miss it's wasted. Like i said, I'm not a Pro, and Ghost is like flash for dummies!

I have been using Garrison a lot more lately, i like it better than Ignite, as it takes one more escape away from the enemy, tower hugging is no longer a way to avoid jax destroying you!!! Also alot of solo queue you dont ever get anyone who will defend. it gives you the option to try to defend once in a while, just hit Garrison stun and laugh :)

Let me explain, no there is too much, let me sum up.

I start with Basic Boots and Pickaxe.

First time i go back I get Ninja Tabi and complete my Guinsoo's Rageblade, hopefully.

Next time i go back i try to get as many parts of the Hextech Gunblade as i can, usually taking Hextech Revolver first and Bilgewater Cutlass if i have enough. Of course if I have enough gold I just complete the whole thing. (Edit: with the hit to gunblade i still like the item, but i get it last, instead of early)

Sanguine Blade- I love the synergy with Jax having so many auto attacks, it works nice.

Atma's Impaler- More Synergy, More AD and HP. Great Jax item.

Warmog's Armor- Really? Noob!!!! Warmog's in Dominion??!?!?! Some people think that just because you dont farm minions its a waste of an item slot. But, you get stacks with Kills and Assists also, not just farming minions. i regularly get 30+ takedowns with Jax in dominions, and you only need 10 to cap the bonus from Warmog's Armor. and more HP just makes Atma's Impaler Stronger, And gives ALOT of HP. Most games don't last this long anyway, but if they do it's because there is too much fighting and not enough capping, if this is the case, you will stack it just fine. try fighting 1v1 against a Jax with nearly 5k hp, see who wins.

SITUATIONAL ITEMS

As I stated earlier, I play mostly solo queue, which means good luck getting good teams every time. In some cases, you have to do everything yourself. When i need to deviate from my build i usually drop my Warmog's Armor and Atma's Impaler. (Edit: gunblade can be dropped now instead of Atma's i think)

If my team has no capturer, or if there is someone playing as a capturer but only trying to get kills (too often this is the case) i will substitute Priscilla's Blessing and a Phantom Dancer, and just do the back dooring myself, still having enough damage output to take down most players 1v1 in the middle if they try to intercept.

If the enemy is AP heavy and fed, like i said, solo queue, it happens....
i go with Wit's End and Lich Bane sometimes Banshee's Veil

The Lightbringer is a good item for those pesky stealthers.

Ionic Spark is fun sometimes. It proc's often with Jax's Attack Speed, nice when Defending 1v2 or 1v3 under your tower.

Frozen Mallet I use sometimes instead of Warmog's Armor The slow is nice, and the 700 HP goes well with Atma's Imapler

Kitae's Bloodrazor helps versus a tank heavy team, stacking HP

Trinity Force is Awesome!!! but it's expensive, i usually don't use it.

Well, that's it. Like i said, I'm not a pro. I'm not looking to get the most Up votes. Vote up, vote down, or not at all i don't mind. Just looking for some feedback, constructive hopefully. I don't care call me a Noob, words on the internet wont hurt my feelings :) If you try my guide and you like it or don't, let me know!