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.
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 a cooldown of 80 seconds. Jax usually doesn't need it in a 1v1. Save it for team fights, where it's really needed.
If used at mid or low health, it can bain enemy and make him over-commit to killing you. It has nice synergy with Maw of Malmortius, getting extra armor and making the item's passive more useful.
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 an early first blood, since you can use it both for dodging and stunning the enemy. It can also help you 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 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.
When reaching level 6, Jax can become a strong lane zoner. Try hit a minion twice and then quickly use Leap Strike into an enemy champion and attack him or her with third attack to proc Grandmaster's Might's passive. You may also activate Empower to maximize your damage output. However, pay attention to your mana pool.
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.
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.
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.
A Blade of the Ruined King strongly amplifies Jax's damage for second output thanks to the item's passive and the built in attack speed which synergizes with his Relentless Assault. It also has a nice active which allows Jax to chase down enemies.
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 level 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 chain vest.
Also consider building Essence Reaver due to Jax's basic attack and on-hit nature of his kit. The items adds cooldown to his kit, the life steal adds some sustain, and it's passive will allow Jax to restore his mana with just basic attacks.
Consider Phage as an early item on Jax. The increase movespeed on hit helps him stick to targets very well, as well as health and attack damage. It also builds into Trinity Force, a strong item on Jax. Advanced techniques of Phage include:
When ganking as a jungler, smite a minion to instantly gain 60 movespeed, you can use this to catch up to your target and use your Leap Strike -> Counter Strike combo to stun the enemy.
When escaping a gank, you can kill or hit a minion to gain 20 or 60 movespeed to get out of that gank. If necessary, you can also Empower -> Leap Strike an enemy minion to both gain some distance and acquire 60 movespeed if you killed it. This can also be used on small jungle monsters.
The Howling Abyss
The Crystal Scar
The Twisted Treeline
Runes and Masteries
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 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).
Try to shut down Jax early in the game because once he hits level 6 and obtains Grandmaster's Might, he becomes increasingly hard to fight 1v1. Ganks from the jungle and zoning him can make him under-farmed and under-leveled, delaying his items and scaling. Although Jax is very strong at late game, he is fairly weak early without sustain.
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.
Most Jax item builds involve him building mostly glass cannon due to the fact that his offensive stats will grant him defensive stats with Grandmaster's Might active. Consider focusing him down first in teamfights.
Remember that Jax is a melee fighter with only one ranged gap closing ability, Leap Strike and a heavy reliance on staying in attack range to keep his passive, Relentless Assault, stacked. Any crowd control to disengage after his Leap Strike or simply staying out of Leap Strike range can nullify his only engage.
Try not to let Jax stack up his Relentless Assault passive before he engages on you. You probably can't compete with 84% attack speed.
Jax mainly combos his gap-closer along with Counter Strike. If you have an escape or a gap-closer on your own, you can use it after he closed the gap with you to escape his stun.
Jax relies on getting close to his foe to land a stun. Try keeping a safe distance for a few seconds after he uses his Counter Strike.
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 not to fight him alone and outnumber him.
When he activates Counter Strike, avoid grouping around or being close to another teammate 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.
Jax is considered to be one of the best late-game duelist and split pusher due to his Relentless Assault, his sustained damage through Grandmaster's Might's passive and Empower, free defensive stats from the said Grandmaster's Might's active. Do not try to 1v1 Jax late-game unless you are certain that you can win the melee trade.
However, Jax's mobility is limited to Leap Strike, so champions with good mobility and/or crowd control can technically "duel" him through short trades and careful kiting over melee "duel" trades. Teemo, Singed, Kennen, Lulu, and Vayne have good mobility and crowd control that will allow them to "duel" Jax through kiting trades.