Runes are mainly used to give you an early game advantage:
AD marks help you do a lot of damage in lane, it makes last hitting easier and harassing more effective.
Lifesteal quints help you sustain in lane before you can afford lifesteal items
Armor seals are needed for trading damage against other AD bruisers.
Flat Magic resists are used to reduce magic damage from champions like Kennen and Vladimir
For mastery I use 21/9/0 page, make sure you have both 10% armor and magic penetrations, it's not worth putting points in critical damage because Jax isn't a critter.
1. Ghost or Flash, you need either one of them for extra mobility in desperate situations; flash is better if you're fighting an ad carry with built in mobility such as Ezreal, ghost is better if you're fighting the likes of Ashe and Miss fortune.
2. Surge is extremely helpful throughout the game, it provides you tons of burst damage in fights; although Jax has a strong damage potential, he needs momentum to gain passive stacks, the 40% bonus attack speed helps you get there much faster, not to mention the large amount of AP it provides.
Because Jax is a melee fighter and he doesn't have much mobility after he jumps in, the least thing you want is not being able to deal enough damage, you need to kill your target as quickly as possible before they could recover and kite you around.
Laning: Jax is a solid jungler, but he benefits more from top lane where he can get farmed. There are several counters to Jax, such as Jayce, Kennen, and Vladimir. When you're fighting these champions make sure you burst them down at level 2 with jump and stun combo, killing them or forcing them to consume potions and play passively. Here is how you do it:
Step 1: Farm passively at level 1 and let the opponent push the wave under your tower. This will reduce your risk of dying.
Step 2: Before you engage, make sure the enemy wave is decimated, early game minions can do a lot of damage.
Step 3: Consume a health potion right before you engage, Use that jump+stun combo and continue to auto attack when they're stunned. They'll lose a chunk of health and be forced to retreat.
Items: Always start with boots+3 pots, you need movement speed to dodge harassment and escape ganks, and you need health potions for early sustain. Boots upgraded to either Ninja Tabi or Mercury treads, depending on your lane opponent.
Early game you should focus on survivability and durability instead damage, because your spells have decent base damage. Here are a list of items you can get for early game:
2 X Doran's blade
Make sure you buy some wards and put them deep in the river, it can prevent a lot of deaths
Mid game you should be getting the recipes of Trinity force and Hextech gunblade, you don't have to rush either one of them, for example you can have a Revolver, Cutlass, and a Phage, benefiting from both life leach and health points.
1. Boots of choice
2. Trinity force
3. Hextech gunblade
After getting your core, you have the option to invest in either AP, AD, hybrid, or on-hit.
For AP you can get:
For AP-based attack speed you can get:
For AD you can get:
For AD-based attack speed you can get:
For Hybrid effects you can get:
For on-hit effects you can get:
There is also a large range of survivability items:
Warmog's armor + Atmas impaler
Quick silver sash
Make sure you have Guardian angel if the game gets super late, the revive passive will force the enemy to kill you twice, kind of like fighting against a 6v5 situation.
Your job in team fight is to jump on the carry and kill him or force him to leave, but if the carry flashes away, and you have no other way to catch up, don't bother to chase him down, just attack the next important (or the weakest) enemy you can put your hands on, do not fall into the trap of chasing one person down and being kited, losing chunks of health, and dying pointlessly. You need to decide quickly whether you can catch up still have enough health to kill the carry, or just ignore him and keep fighting. This is basically the difference between a fighter and an assassin, and you are not Poppy.