Ok so this is a build that I had my best game as Jax with. It was a 3v3 so the result on 5v5 maybe a bit different but I'll try to run it on a 5v5 and make the necessary adjustments.

For the 3v3 I started with counter strike because of the stun, it really helps your team out by holding a fleeing enemy for a couple seconds and with ignite makes it's easy to get first blood for your team. Next is empower because it will be where you get most of your damage from. And next is leap strike, i like to activate both empower and counter strike before using leap strike since empower stacks with leap and if you time counter right you will stun on landing.

Ok so i always start Jax with a Doran's shield on 3v3 just so I can really get in and hurt there team on the first skirmish. You should also pick up health pots as needed. Next you want to build towards either Ninja Tabi or Mercury Treads depending on who is on the other team. Next you'll want to build towards Trinity Force but i showed you how you should build towards it. I took sheen first because I was having mana problems about 5 minutes in, but if you are not having mana troubles then go for Phage first for the extra health. After you finish Trinity Force go for Ionic Spark, your ult's passive makes the 3rd attack do extra magic damage, so on the attack after Ionic will continue that damage with it's chain effect since it hits every 4 attacks, and the massive att speed boost is really nice. I take a Zhonya's then for the armor boost but also for the AP which really buffs empower's damage output. A Rylai's also is a good AP boost and also builds on your "tanky" Jax with Giant's belt. Then sell your Doran's and build your attack and lifesteal with a Bloodthirster to finish your build.