I love Jax, and do really well with him but many people do not, and it seems to me that Jax is one of the easiest champions to play- mostly due to his passive.
Just keep in mind you are an assassin not a tank, and regardless of your health level let the tank initiate and focus on killing their high damage squishies. They should drop in no time. And than mop up the rest of the game once you are feed. Jax just keeps getting better and Rapes. Lots of damage, high health, and life steal to stay alive and keep the health up. The only problem is stuns.

My runes are not set for myself very well because I use them for to many champions and dislike changing which page I have often (I forget more often than not),.. But Ideally-

For Marks get Attack Speed or Armor Pen are my favorite-Pick one and go with it damage is not bad but not great so eh, skip it for something better.

For Seals grab dodge. Dodge and Jax's "E" Counter Strike go amazingly well together as you might assume. Health is an acceptable alternative but Dodge > HP, Jax has plenty of health because of his amazing passive granting a you an plethora of health.

On the Glyphs go for the extra AP per level. The per level bonus is better late game (3x better so play conservatively when level one and two, than destroy everyone once you can) This gives the passive's health bonus and extra damage on spells, or get some magic resist. the magic resist is ok if you anticipate a lot of mages but seeing as you are attempting to predict before the game if they have lots of AP just take what you know will help you and try to kill the mages before they can do much damage to your team.

Quints- I listed the two best options in my opinion-pick what ever you want.
Really I do not care much for runes and use the same set on almost everyone I play

I get the boots and two potions so I can lane longer, return to base to buy when you run out of potions and get better boots and if you can afford it start building sheen or Guinsoo's Rageblade depending on your gold level. Most Jax's like to get the Hextech Gunblade sooner but I like the damage of a BF Sword earlier but if I do not have enough gold for one I grab a Pick Axe and build the Gunblade. Most of my items are interchangeable based on gold quantity at the time you are shopping. Take something so you do not have to return in 30 seconds and waste time running back and forth all the time when you should be leveling or ganking.

I also did not pick a final item, choose what you want. Usually games do not last this long. I like to grab something with one of the following:
[*] Damage (another Bloodthirstier, Black Cleaver, Infinities Edge, etc..)
[*] Attack Speed for the ultimate's passive (Sword of the Divine, Madred's Bloodrazor, Phantom Dancer, etc..)
[*] More durability or a counter for an overly feed enemy champion. This is the best in my mind but it is very situational and if you cant figure out what you need than take one of the first two options.
The final item is left up to you because it is to situational and to be honest at this point in the game you most likely know what is going (we hope) on and would not listen to me anyway. Just make sure to keep using extra gold on Elixir's end game. It can, and has, made a difference for me.

Start with Leap Strike or Empower, I prefer to get leap strike in case it is needed early but try to wait until level 2 to harass or even better level 3 so you can utilize your Empower-Leap Strike Combo than stun everyone near by with Counter Strike and make a smooth exit to the safe bushes nearby avoiding the annoying attackss of those your lane-ing against.
[*] Max out Leap Strike Early for the CDR (Cool Down Reduction) so you can jump around more, this is good for both getting kills/chasing and running away. Jump to minions/allies/wards and hide in safety away from mean people trying to kill you
[*] Grab your ult every time you can, its great.
[*] Next is Empower and finally your Counter Strike. Counter Strike just needs one point early to use, but does not get vastly better, or at least not as much as the others.

Ignite and Exhaust are my favorites but flash is a great move as well. Pick them based on your play style, team, and most importantly the opponents. Against heavy dps teams the Exhaust is great to reduce their damage and also amazing for ganks (I love to gank.) Ignite is needed against Mundo's Ult and nice when someone is running away with low health. Flash is a good escape or chasing tool.

Ghost is ok, Leap Strike can do the same thing for chasing or running away.
Cleanse is ok on heavy CC Teams but I do not use it ever even when I should.

**Don't consider the other Spells**
Revive and Heal are for Noobs, Clairvoyance is for support, Smite the jungler, Fortify the tank, Teleport is not worth it-just run, Rally is rarely used and not your job to have, and if you need Clairity for more mana your doing it wrong somehow, just steal the other teams blue buff and screw with their jungler- They will love you forever for taking it.

Jax is awesome when feed, try to gank and help other lanes, and focus on last hitting when you can. Jax's passive lets him snowball amazingly and after a small lead early on you can ruin games for the other team.

GLHV (Good Luck and Have Fun) and give me good feed back :P
---I am interested in hearing how this works for you mostly because I tend to favor more DPS based characters, and yet everyone else who plays Jax or makes a build for him does more of an AP focus which I do not understand. Attack Speed would for for his ultimate's passive, but AP doesn't seem to scale well enough, doesn't take out turrets nearly as well like not at all, life steal works on his AD not AP.
I never lose to other Jax's though, so I have a lot of faith in my build over everyone else I play which is why I am interesting in how you all like it. Maybe I just play against idiots and Noobs so I rock in comparison.