When laning, Jax is an excellent zoner when in the brush, because his Leap Strike - Empower combo enables heavy harassment with a lot of pressure. Be careful as these two abilities drain a lot of mana.
Jax can use Leap Strike to leap into the fray of a fight or make a quick escape when necessary due to the fact that Jax can Leap Strike towards a friendly unit. You can use Empower to maximize your burst potential when using Leap Strike. Counter Strike can also used right before Leap Strike to lead to an instant stun.
Empower is most effective when used as quickly as possible after a basic attack. This, in addition to granting the bonus magic damage from Empower, effectively doubles Jax's base damage output for that attack. Since it is still another attack, it also counts toward building up Relentless Assault and Grandmaster's Might's passive, so those are accrued faster by using this technique as well.
While engaging with Counter Strike seems to be a safer option, activating it after you take some damage can bait your opponent in focus-firing at you, thus mitigating more damage and may causes your enemies to be over-committed in attempting to kill you.
Activating Grandmaster's Might right before a fight will increase your resistance to both physical and magical damage.
Note that Grandmaster's Might has a rather short duration of 8 seconds and 80 seconds cooldown. Jax usually doesn't need it a in 1v1. Save it for team fights where it's really needed.
If used at mid or low hp, it can bait an enemy and make him over-committed to kill you. It has nice synergy with Maw of Malmortius, getting extra armor and making the item's passive more useful.
Jax can jungle effectively with Relentless Assault and Empower, ganking with Leap Strike and Counter Strike. Counter Strike can be used for killing stronger monsters without losing much health, but you are more vulnerable to ganks when it is on cooldown. Careful if the enemy counter-jungles.
At level 2 Jax is extremely potent at ganking combining Leap Strike with Empower to deal massive damage, although he should continue to jungle until level 3 for the stun from Counter Strike.
In 3v3, there can be a lot of fighting even before minions come. Counter Strike is good for getting early first blood, since you can use it both for dodging and stunning the enemy. It can also help you on solo top, especially against champions who depend on autoattacks. You should max Empower though, you can use it many times in a longer fight and it helps you heavily with jungling and killing Vilemaw.
While playing against a Fiora or Warwick, you can dodge their ultimates with your Counter Strike, it might save you from a gank or team fight.
Be patient with Jax. Often the opposing team will focus you down due to your high damage output. If you are having a bad game, just focus on gaining gold through minion kills, kills and assists, and jungling. Once you get the items you need, you will be much harder to kill.
Be careful in team fights with as Jax is a high priority target for the enemy team due to his high damage output and squishiness. Jax is reliant on his team to support him so make sure that you are on a team with initiation and hard CC and that you are at most the third person to enter the fight, behind your tank and off-tank. Try to save Leap Strike to escape when getting focused, rather than to engage on an opponent.
Even with his fairly high base stats, as a carry Jax is very item dependent for his damage output and survivability, being able to obtain gold from any source and building the best items at the most opportune time is crucial to his mid and late game.
A Sheen with Leap Strike and Empower allows Jax to deal a massive amount of damage in quick succession. Although Leap Strike will not proc Sheen, it will proc Empower and leave Jax close to the enemy and ready to land a normal attack which will in turn proc Sheen. This all happens very fast, making it a good way to harass or burst down a target. If you find this efficient, consider a Lich Bane in a build with more ability power and Trinity Force in other cases.
If you choose to invest strongly in ability power, Lich Bane is a great choice. Empower's cooldown is very low on its higher levels, allowing frequent use of Lich Bane's passive.
Wriggle's Lantern is an effective laning item for Jax. The life steal increases sustainability, the proc is helpful for farming, and the Sight Ward promotes pushing and preventing ganks. Overall, it provides a large amount of lane control.
Guinsoo's Rageblade synergizes with his passive and offers a good source of mixed damage. Consider this item in an offensive hybrid build, and remember than you can start stacking it by using abilities, especially if you engage on an enemy with Counter Strike, Empower, and Leap Strike all at once. In a hybrid build, it is a solid third item after boots and Hextech Gunblade.
The attack speed of Wit's End does not make it an offensive item and is mostly for the magic resistance. As Jax is mostly an offensive champion, Wit's End could probably be replaced with a more useful offensive item.
If you find Jax squishy and are looking for an item that offers both armor, magic resistance, and is rather cheap, consider a Guardian Angel. Once enemies see a Guardian Angel on a Jax, they'll have to really consider whether or not they want to focus him rather than some other squishy target. Guardian Angel should only be considered after completing the core damage build though.
Rylai's Crystal Scepter, besides being 145 gold cheaper, procs on all of Jax's abilities, including the third hit from Grandmaster's Might, making it fairly reliable at slowing enemies down.
Consider Muramana, Phantom Dancer, Hextech Gunblade, Wit's End and Iceborn Gauntlet. The various sources of damage (crit, on hit magic damage (Muramana usually hits for around ~110 at lvl 18's max 1800 mana for 16 autoattacks), sheen procs, gunblade's active) will make for a potent burst, as well as giving good offensive stats that specialize in attack speed (good luck getting the gold required). If you have experience with Olaf, consider playing Jax as a more bursty anti-carry.
This build can be shrunk down to an intermediate level by building vamp scepter, hextech revolver, tear/zeal, sheen, boots and chainvest.
Being that Jax is a mainly offensive champion, he is probably best with offensive masteries. Offensive masteries (21/9/0) are seemly better for his damage and with defensive masteries (9/21/0) there is a noticeable decline in his damage while not providing the survivability that one would expect.
A rune page of attack damage and ability power per level can give a powerful scaling amount of bonus damage on all of his abilities, or a flat attack damage and ability power rune page can give a massive burst of damage on Leap Strike and Empower for laning, leading to a an advantage over the opposition during the early and mid phase of the game. This is especially useful for shorter games, such as those on the Twisted Treeline.
Jax can also make use of attack speed runes effectively as they improve both his physical damage (in basic attacks) and magic damage (Grandmaster's Might's third hit).
Frozen Heart and Randuin's Omen are the only two items that slow attack speed, but they both also have high amounts of armor. Where the latter is generally reserved for tanks, the former is viable for a larger variety of champions, thanks to the CDR and high amount of mana.
Focusing Jax when his Counter Strike goes on cooldown is one of the best ways of getting rid of him. But since he is one of the best duelists in the game due to Relentless Assault and the high AD and AP ratios he has on his single-targeted abilities, try to outnumber him.
When he activates Counter Strike, avoid grouping around when the AoE stun activates.
Keep in mind that Counter Strike only allows him to dodge auto-attack, abilities will still damage him. However, note that Area of Effect abilities are still reduced.