I'd like to start off by saying this is my first build so don't be to harsh, but constructive criticism is definitely appreciated. I'm still tinkering around with the build specifics.

This build is based around armor penetration and lifesteal stacking. The amount of damage and ability power from these items gives a substantial amount of health end game and the amount of lifesteal makes jax last in a fight.

The runes were selected to give maximum armor penetration which is important as I do not use madreds, my build is based around a straight physical damage build. The dodge runes are used so Jax will proc his "E" skill more as well as for general survivability. The attack speed runes are just personal preference because I feel that is one thing this build is a bit shy on

I go with a 21/9/0 build, 21 for the damage obviously, 9 to get down to the dodge speed increase

I start with the Dorans shield because I feel the extra HP and health regen are important early game, also a potion for early game so you can stay in the lane a little longer and farm up more.

Next you get the Ninja Tabi boots for the dodge which benefits both your counter skill and your ult.

After that you get guinsoos blade which is where the snowball effect comes in. Its based around the concept that once Jax gets rolling with a guinsoos and his ult attack speed boosts he just gets beefier and beefier. Basically this works well because it benefits his damage, his skills, his health bar, and his attack speed.

With all that attack speed you need some damage behind it so I felt the black cleaver was a strong option. The armor penetration helps keep tanks from countering you combod with the runes, and if the tank wants to counter with bulk armor he will have to neglect magic resist which is one thing this build aims to do.

Next I get a starks because it has life steal, more armor penetration, and attack speed

I finish it off with 2 blood thirsters (sell the shield for the space for the 2nd blood thirster). The high damage output when they're filled up + the life steal makes it so jax gets a massive hp boost and can life steal tank

I select empower because it is your most spamable skill and I usually lane with a stun/slow partner so I let them use theirs on the runner and i can get that extra damage while i exhaust them early game. Leap attack is a 1 point wonder so I do that 2nd, and then I alternate empower (most spamable/useful skill) and counter, getting my ult when i can.

I select exhaust and flash for a few reasons. Exhaust is very useful as a slow because Jax lacks in slows/snares.

Flash and Ghost are interchangable, I like flash because it allows me to get out of sticky situations a bit quicker but ghost is better for chasing runners so you decide

For creeping early game I like to hit the empower button and last hit creeps with that, last hitting otherwise too but empower allows u to time it a bit more loosely which is good if your opponents are waiting to spam stuff at you and try and predict your creep farming patterns, it's mana efficient and you get in and out, the dorans allows you to heal enough to recover from enemy harassment

Overall this build provides high damage output with moderate survivability especially if played with the proper team, a solid jax can be a good late game carry.