Ah, yes. This is the NoobyJax build. It's my first build on here.

Basically this is how I play Jax, and personally I find it pretty much self-explanatory, the reason I get Atma's is because it gives a nice crit chance, which is always nice + extra survivability, something that Jax needs while getting the procs for his ult and guinsoo's.

The item build is pretty common for Jax, attack speed and damage benefits him greatly. Guinsoo's is for an obligatory item for him since it goes hand in hand with his ult and gets your atk speed up to over 2 pr sec, this is where you'll really see why life steal is amazing on him.
If you ever get as far as Bloodthirster, you should be able to solo Baron without dropping too much health.

I always get Jump Strike first, it's a great harassment skill, and combined with empower at level 2 you should be able to land a few quick kills early if you don't face a too hard lane and your lane partner actually does something. The combination of empower, jump strike and ignite drops most champs even at level 2 or 3, if you want to make sure you get the kill, get some creep aggro so your stun comes up, then empower, jump stun and it's gg.
I choose to max out Empower first since it's Jax's main damaging skill, then I go for the jump since the combination of the two can drop most champs pretty quick even late game. The stun is what I max out last because even if it's just 1 or 2, it's still a stun and though the damage output isn't too high, you should have no problem dropping the enemy.

I used to go Flash and Exhaust, but I find it better to use Ignite instead of Exhaust because of the extra damage that might be the final touch you need to drop someone. Flash on Jax is pretty good for both chasing and running away, you'll already have pretty good control over the enemies, but if you need a quick get away from a teamfight or someone runs away, flash is really what decides life or death, you're Jax, you can't take on an entire team, so escape mechanisms can be really helpful thus allowing you to change this with Ghost which works pretty much the same way. It's really up to you.