Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Jax is pretty versitile when it comes to runes. I like dodge cos u get extra stuns and less damage. Attack speed marks for that extra bit of attack speed to get your ult started up, and Magic resist glyphs to get less damage from spells.
Leap Strike: This is a good initiator, chasing mechanism, running away tool, and if you get really lucky you can leapstrik onto an minion over a wall. I generally try not to use it as an initiator unless I need to because it has a long-ish cooldown (compared to his other skills early on) and I then use it to chase. It works excellent with Jax's two other skills to cause some havoc.
Empower: Empower is a wonderful skill to put out some extra damage in teamfights, get a last hit on a minion (can no longer hit several of the lil buggers >_<;), or to harass. In my build I try to get Counter Strike maxed but if they mostly have people who won't be dodging your attacks then it is usually worth it to max Empower first. Always Empower before a Leap Strike for even more extra damage.
Counter Strike: Jax's best skill besides his ultimate. The dodge chance gives you wonderful survivability. The stun has several uses. In a close team fight it can stun all of you enemies giving you and your allies the upper hand. It can be used to out damage another physical DPS in a 1v1. It can be used to stop a fleeing opponent long enough for the last three smacks to kill them. You just have to feel when the stun is needed.
Relentless Assault: This is why Jax is Jax (and now his ultimate is even better :D). This enables Jax to get his attack speed to max really, really fast. Then if max attack speed wasn't enough every third hit (wich occurs almost once every second at max) hits them harder.
Well, those 2 summoner spells are great to chase enemys, or to run, instead of Ghost spell, you can also buy Flash, its pretty good too.
Leap-Empower: This will be your main combo from lvl 3 onwards. It allows you to jump on someone and deal a ton of damage with just one move, at the beggining, use leap-empower and run, your enemy would leave to recover or you will kill him
Early Game: Start out with your Doran's Shield and a Health pot. head down your chosen lane. Try to get a lane with someone who can disable, or harass with you. Supports are always nice too. Focus your attention on farming last hits, and once you hit lvl 3 just start doing leap-empower combos on the squishiest enemy, hopefully forcing them to go back. Once you hit level six, farm like a boss and really go for kills.The first time you go back you should have a kill or two, or at least enough gold to buy full boots and a blasting wand. If you only have enough for boots, pick up a ward or two. Put them in strategic places, by buffs, or if you want to get fancy, in places you could leap too as a get away, like in the bushes by the enemy tower.
Mid-Game: Lane as long as you want, but as soon as team action starts, go right to your team. Try and run around ganking, and in team fights always focus the squishy. Leap-empower in then run out, or keep beating on them if there team allows it. Focus on getting enough money for Guinsoo's and hextech, and go for lizard and golem whenever they're up, especially the enemies.
Late Game: At this point you should have most of your items. Focus on wrecking the squishies in fights, and stunning whenever you can. Put up wards at buffs, and wait to see if you can gank enemies there.