Jax is a champion that has his own odd little niche. He is a tank/dps/ad/ap damage dealer. Not many champions fit all these rolls but Jax just it very nicely if you can find a good lane for him to solo.

With the change in meta to tanky top lane solos Jax has found a new home top lane vs these match ups. Jax is an amazing counter to lane udyr and lane Warwick. He beats both in lane and is stronger end game.

For runes I run dodge yellows, magic resist blues, flat ad or
Scaling ad quints and armor pen or ad reds.

The armor pen and ad rune differences are not to much so you
Can decide which you like better.

I run 21/9/0
I don't pick up the increased crit
Danae mainly because I don't ever get crit items before the game ends.

As for the 9 defense nimbleness is just way to good to pass up.

I start with armor 5 pots.

I try to stay in lane until I can grab wriggles plus boots.
I only go dodge boots on jax if the enemy team only has one hard cc. If they have anymore than that stick to the mercs.

I build the pick axe portion of gunblade first due to the slow being amazing for following up leap strike combos. I finish hextech and
Grab rage blade.

Once you have these items your
Core is done and anything else is luxury. If you are having trouble living then build tanky if not then grab dps items.

I grab one point in q w and e then start maxing q and w. Maxing w first will let you trade better in lane but makes you worse at chasing so you can decide which path you like better.

I max e last it is a one point wonder.

Refer to the link below for jax skill info and other things.
http://leagueoflegends.wikia.com/wiki/Jax

Jax's main mind set in lane is to just farm
Until you can get wriggles and Boots so focus on that cs!
Once you have wriggles and boots you can go far more aggressive.

Vs melée walk up autoattack w then auto again to harass. Vs range press w then leap strike to the enemy and then run back.

If you are ganked don't forget you can Leapstrike to your own minions toGet away.

You can drop your wriggles ward over a wall then leapstrikr to it. This is Useful for escaping in The jungle.

Before engaging on some one auto attack minions to build up you ultimates stacks. This way you can go crazy on them when you leapstrike to them.

Once Jax has hex and wriggles you can usually 2 man baron so drop a pink ward on baron and grab an early baron.

Jax can go all in and tunnel vision a squishy, but most good players will get a way.

I like to empower leapstrike combo a squishy then run away until my combo comes up again. Jax is a great Skirmisher so play him like one.