Jax is incredibly strong in lane, if you have like, at least 3 fingers, a stump to move your mouse with, and at least a foot pedal to move and attack with.

I highly recommend this video by LoL Dobby on how this superserver Challenger plays Jax.

Here's the basics.

Versus melee, you're very strong level 1/2/3. Short trades are ideal; a single E with full bonus from blocking minion autos does 110 + 8% max hp. It's very easy to get a few auto attacks and a 50 damage W off during that time, meaning you can easily chunk your opponent for 250+ hp.

This doesn't mean you can't take an extended fight, as you're very strong in those too, especially with Lethal Tempo and your passive.

If you can't win a fight, just farm. You still scale well.

Against ranged champions, with Grasp of the Undying, remember to put at least three points into Q. Your trade pattern becomes stack grasp -> Q -> auto(grasp). Do this off cooldown, and suddenly you have a ridiculous HP advantage, and can force a flash or a kill.

Before your first item, if you don't know what to buy, or what you're going to buy later, buy a Caulfield's Warhammer. It'll get used eventually.

If you want to win, you'll be going Divine Sunderer anyway, so build either a Sheen + Long Sword or Caulfield's Warhammer.

My favorite build is Divine Sunderer, Plated Steelcaps (also known as Tabis), Blade of the Ruined King, Banshee's Veil, Zhonya's Hourglass, and Death's Dance. This build uses Jax's new heavy AP scalings to gain resists and damage, at the cost of having less AD overall. However, the AP more than makes up for it.

You can do stuff like, Flash -> Q -> E -> Zhonya's to engage a teamfight with a big AoE stun.

Be careful though, since Grandmaster At Arms's resistances ONLY SCALE OFF AD. Thank Riot, they gave us a rework that makes AP Jax seem viable, except he doesn't get any resists anymore, so we don't get to have fun with Nashor's Tooth or Riftmaker.