PLEASE READ AND TRY THE BUILD OUT BEFORE VOTING. THANKS.

My first build ever so go a little easy on me.

Jax is a very complex champion.
Play him right on 3v3's and it's GG every game.

Personally the way I build Jax is very different every game.
A good Jax needs to learn to analyze the enemy team and to build accordingly.
For example, if they are all Physical, don't build MR..

But things like that should be common sense, even to the newest of players.

Runes are easy to choose but I usually go with two rune sets.

9x Magic Pen Marks
9x Dodge Yellows
5x Flat AP Blues
4x AP per level Blues
3x Movement Speed Quints.

These are the best AP quints you can possibly have on Jax.
My other rune set is :

9x Flat Damage Marks
9x Dodge Yellows
9x Flat AP Blues
3x Flat Dmg Quints

Since you're gonna be building a TriForce on the Dmg build you really don't need the extra movement speed.
Even though, The extra 7.75 Damage makes up for it by a long shot.

The cheapest move in the game.

W -> Q
Thats it.
Empower your staff then jump on them for insane burst output. Not to mention you get a free hit after you land on them.
What's not to love?

Flash
Ignite
Ghost
Exhaust

Any other Summoner Spells are a waste of space.
Find two you are most comfortable with and stick with it.

I personally use Flash and Ignite for 3's, because I mostly play three's anyway.

NEVER INITIATE AS JAX

I can't stress this enough, I see bad Jax's all the time initiate and team fights "OHH I HAVE DODGE I'LL SURVIVE DONT WORRY"

And then they die and cost the team an ace because their team has no damage output.

Play like you're an ashe.
You're a clean-up carry.

Go in, AoE stun everyone, W -> Q on the weakest squishy then get the hell out of there.

Farming is much easier with Doran's Blade or Doran's Ring.
I vary between the two based on the enemy team.

If the enemy team is Heavy DPS/Physical, you're best bet is to go Doran's Shield.

As Jax you shouldn't have any mana problems especially with a Doran's Ring. Don't hesitate to jump on a minion to last hit, it's worth the 23 or so gold for some measly mana problems.
Also as Jax, you may tend to push farther than you intended.

To stop this dont auto attack at all and watch the minions health until your damage is more than the minions current health. (Last Hitting)

This way you are farming and zoning at the same time.

Be smart, Jax is a tad OP but you'll get used to the
"OMG urR b4d p1ay a gud ch4mp u nub"

You have a spammable stun, USE IT!

You're not invincible with dodge so play efficiently and always remember:

"Imagine if I had a real weapon"
;)

And thanks for reading all of this.
Enjoy!