I play Jax a lot, and I'm probably not the best player, but most of mistakes I make are due to errors in judgment, not my build of Jax. This is my first guide, and I'd love feedback not related to flaming. In this guide, I'll present a few branching ways to build Jax that I use situationally. Much credit goes to I Play Jax's Jax guide, which basically got me started on playing this character well.
So here goes...
Section 1: Skills
I'm going to ignore the description of each skill, and just discuss what each skill is good for.
Empower-- I don't use this skill early game very much, but I've seen other people play with it. It seems a better mid-late game chasing skill to get the last extra bit of killing damage.
Leap Strike-- This is by far my favorite skill that Jax has. Early, it's fantastic for harassment, and even for killing if you manage to draw a dumb lane opponent. It also works fantastically for chasing. It scales with AP, but I've never found a great way to build an AP Jax, so I'll leave that for other people to discuss. You want to max this skill out as soon as possible, along with Counterattack for purposes of early-mid hero kills.
Counterattack-- Also a great skill, and incredibly useful for all Jax builds. Getting a dodge and getting a counterattack is a great way to ensure that you'll get a few more hits in on your opponent, especially early. Maxing this out in tandem with Leap Strike early will ensure that you can take down fleeing heroes and squishy ones.
Relentless Assault-- Win. Take it at every possible opportunity.
Here's the build I use:
1. Leap Strike
4. Leap Strike
6. Relentless Assault
7. Leap Strike
9. Leap Strike
11. Relentless Assault
12. Leap Strike (Max)
13. Counterattack (Max)
16. Relentless Assault
18. Empower (Max)
I'm not sure if these levels actually work properly, but you get the idea, even otherwise.
Section 2: Masteries
I go 0/21/9 for the dodge bonus. I always feel like Jax's role as a late-game carry means that he should get the +exp from the Utility Tree. If you do this along with the +exp runes that I'll discuss, you should be grabbing hefty extra experience (+11%, I believe) and you'll be slightly ahead of whoever is laning against you. This will also allow you to jungle buffs effectively earlier.
Section 3: Runes
You want to stack attack speed and dodge runes for all areas but the Purple. This will allow Jax to get the hell out of the way a little bit more. Since he's squishy, this is exceptionally important in terms of survivability. Take Experience runes with your Purples. This will get you a slight step up on your lane opponent, allowing you to **** him up easier. Then you can get gold and keep building what you need.
Section 4: Items
This is where I'm going to branch out a little bit. I *always* start with Vampiric Scepter. My build usually goes like this:
Berserker Greaves (Discussion to follow)
Vampiric Scepter--->Malady (Buy Daggers along the way if necessary, but you want to get this item for its +AS, Lifesteal, and anti-hero debuff)
Phantom Dancer (As much as you might want to, DON'T RUSH THIS FIRST. You want to make sure that you have the +AS already, as well as the Malady, for reasons just described)
I start all games with these items, in that order. From here, it can go in a few different directions. Sometimes, I'll take the Rageblade because of it's stacking AS and AP buffs. The AP works well with the proc on Relentless Assault and the AS works well with Relentless Assault, as well. I'll also take Phage--->Frozen Mallet if I'm having troubles chasing or have survivability problems. Ultimately, you will want an Infinity Edge if you are doing well enough to grab it. It's a beatstick and will help you kill anybody, and even *every*body. Bloodthirster and Black Cleaver help beat some respect into pesky tanks. Banshee Veil will allow you some survivability against casters. Ryze and Kassadin are the heroes most likely to elicit your purchase of the Veil. If you're doing really well, just start stacking Infinity Edges, and you'll win.
I don't take Ninja Tabai because I feel that the extra attack speed helps Jax farm early, and allows him to wreak more havoc on the person laning opposite him. Ninja Tabai does help with Dodge, yes, but Counterattack spamming is crippling to your low early-game mana pool and will leave you out of mana pdq. Even Leap Strike will do this, but the lengthy early cooldown on Leap Strike will discourage you from abusing your tiny mana pool to use it.
Feedback is not just welcome, it's solicited for this section.
Section 5: Summoner Spells
I generally go with Heal/Teleport because it allows me to clean up messes that my teammates cause, but when I'm running with a premade, I will go Smite/Flash to allow for better jungling and good gold in the early game when you're making your item build.
Section 6: Gameplay (The important part)
Early Game (Levels 1-6): Get solo as often as you can. You'll have some dip**** Ashe on your team who thinks she is entitled to solo mid, but she would be wrong. Call it and get there. Don't let anybody stop you. This is important because it allows you to get more experience and gold early, and get cracking on your build. Jax is a monster if he outlevels somebody and ganks him/her. When in your lane, use Smite to last hit creeps you'd otherwise not be able to grab, especially cannon minions. Focus on last hitting, and occasionally harassing the opposing hero. If the hero gets too cocky, grab a Counterattack by running into a few creeps when he's not looking, and Leap Strike him, and immediately Counterattack. Use the few seconds to get in a few meaningful hits, then get back to behind your creeps. Use good discretion in doing this, as you don't want to get yanked into a turret by a Blitzcrank or rooted and nuked by a Ryze. Don't die early, if only because you're not getting gold at a reasonable rate while you're dead. If you're playing this well, you may be able to get Malady by level 6.
Mid Game (Levels 7-14): At this point, you should continue on your build, but consider helping other lanes. A well planned gank should net you 2 kills, and all the gold that comes from those kills. People in other lanes should be at a lower level than Jax, and you should be able to obliterate them. Use the Leap Strike+Counterattack combo to cut down escaping enemies. If you need it, grab the Lizard Buff at about level 10 and use the slow on that to make sure people don't get away. Once you've killed the heroes in the lane of your choosing, take down the tower, and return to your responsibilities in mid. You'll want to continue this ganking thing for a while, since it'll net you good gold, and it's fairly low-risk if you manage to use some caution.
Late Game (Levels 15-18): You're a monster. Don't run into 3v1s (usually) or 4v1s, but you should be able to take most 2v1s, especially if you've garnered your Infinity Edge, or built up your Bloodthirster. Remember at all times that you're squishy, and that a well placed spell or two can cause you a lot of pain. When you get into a fight, try for as much sustained combat as possible in order to build up your Relentless Assault. If you've built the build I've suggested, your attack speed will cap out at 2.5 attacks/sec. This is what you want. People will have no idea what is happening to them until it's too late. You can also use Counterattack to disrupt channeling spells, like Nunu's ultimate and Fiddlesticks's Crowstorm.
Feedback is also welcome in this department.
Section 7: People you don't want to lane against
There are a few heroes that work as solid counters to Jax.
Ryze-- This creepy mother****er is not somebody you want to be caught alone with. If you don't harass him properly, or exercise a LOT of caution, he'll root you and nuke you to hell. This is a bad thing. Using caution can usually make this a moot point, though.
Heimerdinger-- Who *does* like laning against this guy? He's terrible for Jax because Jax has to walk into his turret nest to cause him any damage. This is hard to do when you're squishy and melee. Ask for help, or switch lanes if Heimerdinger shows up.
Ashe-- This is very doable, especially if you catch Ashe in a weak moment and jump on her/stun her. I hate how she kites, though, and you'll hate it, too. Make sure that you're harassing her properly and making her think twice about committing to attacking you. Once you get Relentless Assault, though, she's a lot easier to obliterate.
Kassadin-- Kassadin's silence is annoying, and you'll find considerable trouble dealing with him mid and late game if he's built properly. The problem with the silence is that it prevents you from harassing him effectively with Leap Strike. I haven't found any great ways to deal with this other than keeping him at bay enough that he won't farm gold. Wit's End could also help, instead of Last Whisper, in terms of keeping him without his enormous mana supply that he will need to drop the mid and late game nukes on you.
So this has been my first guide. As I've said throughout, I'd love feedback and ways to improve. I'll also answer questions as well as I can. Thanks for reading, and good luck with the master of the beatstick lamppost-- JAX!