A Guide to Jax
Hi there. Are you looking to learn how to play the grandmaster himself? Here we learn to imagine what life would be like if we had a real weapon. 
Quick background - I've been playing Jax since season 2 and have mained him since S3. Since then I have been Plat-Dia, peaking at Diamond 1 last season, though I have been playing on a less "tryhard" basis since then. I had the most mastery points (at 1.9M) in EUW at one point this season (which was about the same as Hash at the time).
Anyway: Keystone? Your 4 options, in order of likelihood are: Conqueror PTA Lethal Tempo Gliacial Augment
Conqueror is the one that fits all. This is the default because it is so damn strong. As Jax you don't build pen. That makes conqueror so cost efficient for you. As Jax you have tons of attack speed. The bonus AD from Conqueror is superb for you. Your mindset and thinking should be - when are the other choices actually better than conqueror?
PTA - this is a good duelling keystone like Conqueror but doesn't scale as well. It is better against squishy targets, especially in the early-mid. Take this if you are against someone who isn't going to build resistances and if the enemy team doesn't have any tanks. This is also a good one for Jungle Jax, as you won't get as much opportunity as Conqueror to stack it beforehand.
Lethal Tempo - This seems troll but can actually be insane for Jax. Why? If the enemy team has little/no CC, this is the one to get. With this you can literally pop off on 3.8 attack speed with a suitable build. It can also give you surprising all-in power in the early levels, which can bee good against melee champs. Again, this can be a good choice for Jungle Jax.
Glacial - UWOT? Yeah, it can be good. HOW? Well, some matchups for Jax can be rough because he can't stick to his lane opponent early. 
These champions can usually stomp Jax or at the very least go even - so with Glacial you can leap onto them, stun them and get a couple of auto's off. After the stun they will then be on the heavier end of the slow so you can still stick to them. And if you maxed your Q, you will be able to q to them AGAIN. It can work - but comes at the expense of some late game power so it is a trade-off.
Laning.
Your role here is to get that juicy CS. Your lane opponent is an inconvenience. The only thing stopping you from getting every. Single. Last. Minion.
You are very strong level 1 vs melee champs. You are strong trading at level 6. Between those levels, on the whole, you are not strong. Try not to go rambo. Do what you need to do to get every last minion. If you get good CS, inevitably you will win the lane. By default. It is down to your opponent to try to stop you doing this. And often times they will try beyond their means because they know this too. Start with 
in nearly every situation. Jax is mana heavy in the early levels and has no natural sustain so this is superb for him.
My biggest advice in lane is: don't waste your E. Jax is very vulnerable without his E. If you used it to engage and your opponent did the sensible thing by walking away, do not hang around unless you know you're in a good spot. Naturally you are in trouble here, so look to go into brush or just lay low until it's coming off CD again. Max your W in most circumstances. Max your Q if you know you out-DPS your opponent and just need that sticking power (usually vs mages).
Items.
Jax's core item. Not building this is either troll or a niche build. Jax is among the best users of Tri Force and uses every stat it gives.
This is superb for Jax as his attack speed gets a lot of value out of the passive. The active also lets you reset your autos, and couple this with your W means rapid fire, hard hitting autos for good burst and eventual DPS.
Go this or the red smite as Jungle Jax. No harm in going cinder, It's a solid pick too. For soloQ I think this is better.
I absolutely love this item on Jax. It makes you twice as strong. Coupled with Triforce this will let you do insane damage, while being very mobile and actually quite tanky thanks to very low downtime on your E.
I'm not going to go onto all items, but there's a quick summary of his best items. What should a full build look like?
- The last item is dependant on the match. This could be anything from BORK to Randuins
What is the strongest carry build possible on Jax? Easy:
- seems like you'd be too squishy, but actually your ult will provide you with insane amounts of resistances, and couple this with the healing from Gunblade and your insane damage and damage reduction (your E), you will be able to pop off and destroy anyone in the game in a duel. I propose this build on Jax is the strongest late game fighting build possible in the entire game (you could trade out merc treads for phantom dancer or bork of course but not in a 5v5). I have literally oneshot late game fiora with this without taking any damage. And solo Baron.
TLDR: Don't feed early, build offensive items in SoloQ to carry, destroy champions, monsters and towers with easy late-game. Be the powerhouse you were always meant to be!
