Jaximus Maximus. Lame name I know.

What I think of Jax. After playing him several times, I figured out quickly how to build him. A lot of people say hybrid. I can see it working, everything can work, I've made an AD Lux and gone legendary many times. It's just a matter of what works BEST. I prefer my jax NOT to be a one-jump-Lich-Banes sort of guy. I like hitting things. So a DPS/AP build is the way I got. Simply because his ult provides the AP in case someone is all Armor.

Self-Explanatory. Defense, Dodge, and CD reduction on Cleanse and Ghost. The main problem I noticed is when I dodge, it is usually followed by someone stunning me. I like to fix that. Plus ghost is nice for chasing those pesky stragglers down.

These are my Runepages pretty much. One is all AD, one is all AP, one is all Armor/Magic Resist, and the last one is all Energy Regen. If I am going to make a Runepage for something, I make it ALL of it that something. Who doesn't like + 55 AD at lvl 18?

I prioritize the dodge chance. Now I know most people call me an idiot for it, but his only disable is that AOE stun. And he is supposed to be an anti-DPS DPS. Plus, it helps out in the lane-ing phase a ton.

Trinity Force. Enough said. This I found is the number one late game item for Jax. Movement speed, attack speed, Sheen, Phage, mana, I could go on and on. It's made for him. It has his name on it. Ninja Tabi for dodge chance, Starks Fervor for Attack Speed and Lifesteal, and the rest are self explanatory. With a build, up to the Rageblade, I solo'd a fed Tryndamere, Wukong, and Lee sin. And that includes all their exausts, ignites, and that damn kick. A fed Jax is an anti DPS DPS Jax. Late game you murder. Everything.

Play it safe early on, farm as much as you can. Jungle for buffs to help with your mana/catching problems. Try not to initiate team-fights unless you feel you can survive. (You know what I mean). Focus the DPS char's. (The ones that do damage, not talking about Pantheon here LOL). This build works for me, and it should for you.
As proof (And of the solo-triple kill) :
FYI- That loss was because our Rammus quit. And also, as a sidenote, I was a Hybrid Annie in the other match, 250 AD and 500 AP. It was tricky, but fun.