Welcome to my guide on Jax! It's not really that much different from most guides you'll see on this site, but I'd like to highlight the various itemizing options Jax has, as well as nuances of play and such.

Jax is a strong solo top, and a great champion to deter the tanky DPS metagame. His burst is extremely strong, and his cooldowns are rather short, so take advantage of both those facts in combat.

In this guide, I'll explain the two builds I use for him; pure DPS and tanky DPS.

Pros:
-High burst damage once you've assembled Hextech Gunblade
-Strong sustain
-Very strong snowball potential, gets absurdly strong when fed (I've actually been able to 1v4 and kill all of them, albeit when Jax was QQ OP a month ago)
-Can 2-man Baron Nashor with your ranged DPS once you've assembled Hextech Gunblade and Guinsoo's Rageblade
-Counters tanky DPS quite well as he gets frighteningly strong the more he battles
- Counter Strike procs off minion dodges, which is hilarious but almost unfair when used effectively
-He duels extremely effectively, and even counters enemy duelists like a Xin Zhao, Tryndamere or Nocturne
- Leap Strike can target allied units, giving him incredibly strong mobility

Cons:
-His combo is countered by Banshee's Veil, as the proc will block all the damage from your Empower + Leap Strike combo
-His mana is awful early and doesn't get all that much better late
-His farming ability is decent at best before level 6, making his early laning phase pretty weak
-He's going through the grinder of the nerf bat (See Irelia, Jarvan IV, and most recently Gangplank) after being the topic of QQ OP on the main forums, so he's losing viability with each nerf
-He's prone to being kited
-Mages with high range such as Xerath or Lux pretty much poop on him, because he relies on Leap Strike to get him into melee range, and while Relentless Assault's active gives him a fair amount of Magic Resist, it's faily negligible to enemies who build Void Staff

Because I build Jax in two different styles - DPS and tanky DPS - I'll highlight my setup for both.

FOR THE DPS BUILD:
For marks, I like taking Greater Mark of Attack Damage. The ratios on Leap Strike and Empower are both very high, so the bonus attack damage really helps bolster his damage output. I've yet to decide if they're better than Greater Mark of Desolation, but it's a matter of personal preference, I suppose.

Greater Seal of Evasion because you pretty much bought these seals just for Jax. If you don't own them, Greater Seal of Scaling Mana Regeneration is an acceptable option which helps Jax's mana problems.

Greater Seal of Scaling Ability Power again because of how well his combo scales as you build damage.

Greater Quintessence of Movement Speed is a personal preference; I like the mobility mostly because of the fact that I dislike having to burn Flash to chase down and combo kill my opponents, but if you would prefer, you could take Greater Quintessence of Desolation, Greater Quintessence of Ability Power, Greater Quintessence of Evasion or Greater Quintessence of Health. Jax is a flexible character when it comes to runes, and you can work with whatever setup you feel is best.

FOR THE TANKY DPS BUILD:
Greater Mark of Desolation are used because you aren't taking 21 mastery points in offense, so you need the armor penetration to help your damage not be awful.

Greater Seal of Evasion is still, in my opinion, the only real option for Jax. If you want an alternate, tankier option, Greater Seal of Resillience helps him survive most solo top and most jungle ganks.

Greater Glyph of Scaling Magic Resist is a cheap set because they're 205 IP a pop, and they're probably the best defensive glyph in the game.

Greater Quintessence of Health are by far the best survival quints out there. Greater Quintessence of Evasion is an option because it empowers Relentless Assault's active, but it's up to you.

FOR THE DPS BUILD:
The DPS build focuses on upfront burst and insane damage output, but leaves you pretty cold on the survival end of things. You have a fair amount of HP, yes, but Kog'Maw and Vayne pretty much walk all over you.

I'm not quite sure if Lethality is the best talent for him, but you build Atma's Impaler, so I figured why not? If you want an alternative, put those points into Alacrity instead.

Generally it's pretty simple; take all upfront damage amplifications and anti-defense talents, and grab Nimbleness in defense.

FOR THE TANKY DPS BUILD:
Whereas the offense tree gives Jax the ability to punch through defenses and hit like a truck, the defense tree offers two especially tempting talents for Jax: Strength of Spirit and Ardor . The former helps his sustain, especially after building Banshee's Veil, and the latter is just all-around a great talent for tanky DPS.

Jax is best-known for his stellar use of Ninja Tabi, Hextech Gunblade and Guinsoo's Rageblade. Those are his core items.

I've been flirting with the starting item. While Doran's items are strong (especially on item-independent champions), Jax is not one of those champions who I feel can afford to be buying them. A survivability item is necessary, and I've come to the conclusion that Cloth Armor with 5 Health Potions is probably the best start for him. Most of the time, you'll either fight an enemy AD solo top or an AD jungler. The Armor will help mitigate their damage output. Also, the Armor builds into Ninja Tabi, which you want to build right away; the early mobility helps you land Leap Strike and it empowers both Counter Strike and Relentless Assault.

Jax's main item is Hextech Gunblade, but there is a lot of discrepancy on what order to build the item in. I've always found that maximizing the sustain is paramount, so always pick up the Vampiric Scepter component first. Afterwards, go for the Hextech Revolver and then from there build the Gunblade itself. The combination of Vampiric Scepter and Hextech Revolver both keep you healthy and scale your combo.

After that, go for Guinsoo's Rageblade. As someone who does a lot of autoattacking, I feel that the Pickaxe is the first component of it you build, but if you go back to base and only have 860 gold, pick up the Blasting Wand instead.

After you've assembled his core, you work with your rune and mastery setup and build your last few items. I've built offense on defense setup and vice versa, but the rune and mastery setup compliments the builds they're setup for quite well, imo.

FOR THE DPS BUILD:
Lich Bane helps give Jax's autoattacks a lot more punch. Given his short cooldowns, you'll be able to get a fair amount of mileage with this item. The bonus MR and movement speed are just icing on the cake! (Though admittedly, the movement is why I like building this after the core; landing Leap Strike is paramount to your success.)

Atma's Impaler is just stupidly strong on Jax, in my opinion. It's perhaps the best item that synergizes with his passive, and it helps add to his bulk with the bonus armor. Given how stupidly high your attack speed becomes late game, the bonus crit chance really does matter, as well.

Finally, Black Cleaver is a really strong on-hit item I like buying on him to further help him take down enemy tanky DPS. It adds to his already high attack speed, and helps mitigate enemy Atma's Impalers, Thornmails and the like.

FOR THE TANKY DPS BUILD:
After the core, you want to start building survival. Your damage is still extremely high in the midgame, but you become considerably harder to kill late.

Start with every hyper-carry's dream defense item, Guardian Angel. This helps you keep killing sprees alive, and allows you to initiate team fights with near impunity.

On Jax, a good MR item tends to be Wit's End. The bonus attack speed helps his damage output, and the MR assists one of Jax's biggest problem; casters.

Finally, Banshee's Veil combined with Guardian Angel is pretty much unfair. They have to pop your bubble AND kill you twice! How will they do it?

However, if a specific enemy is doing well, keep in mind that there's usually a defensive item out there to help counter them. Randuin's Omen, while good at providing defense, is deceptively strong on offense as well, providing an AoE slow for you to camp on opponents with or to simply nerf a fast enemy attacker with. Thornmail is a good option vs. a Tryndamere, Xin Zhao, Udyr or Nocturne, and Force of Nature can deter an enemy Cassiopeia or Singed who's doing well. Assess the enemy team and prepare accordingly.

While Empower scales damage and manages your mana better as you rank the skill, shortening the cooldown on Leap Strike is far more important. You want to always have this or Flash up to escape a gank, because you can use Leap Strike into an oncoming allied (or enemy, in some situations) minion wave so that you can create distance between you and your opponents.

Regardless, after you're done maxing Leap Strike, max Empower next to maximize your damage output. Level Counter Strike last, making sure to take a point in Relentless Assault at levels 6, 11 and 16.

I've flirted with these quite a bit, having rocked Teleport, Ignite and Ghost, but my favorite to pair with the ubiquitous Flash has to be Exhaust. A ranged nerf spell helps you not only catch up to, but escape from enemies. It's a must on Jax, in my opinion. If you're worried about the lack of damage output you have and want to secure your kills in lane, take Ignite.

But don't let Leap Strike's blinking ability fool you; always take Flash.

This little subsection is simply a small reminder of things. Jax is farm-dependent to get to that Hextech Gunblade as soon as possible, and getting very farmed will accomplish that. A few footnotes on farming:

-Melee minions take two shots from your tower before being in kill range of one autoattack from like 90% of champions.
- Empower only costs 20 mana, and resets your autoattack timer. Use it after landing a basic attack to increase your damage output. (This applies mostly to minions, but you get the idea.)
-If you're autoattacking a minion to death, and you don't land your Relentless Assault proc in time, it will be saved for the next autoattack you launch. Keep this in mind and time your attacks with your minion wave appropriately.
-You lose your Relentless Assault and Guinsoo's Rageblade stacks if you don't stay in combat. Use minions to keep them fully stacked.
-If you're not doing well, farm an empty lane. Your items can snowball you out of a bad situation. I've gone 1/7 in lane and ended the game 11/22 with a win because I farmed and got two free towers while my team battled a bunch. I like to tell people, for those games where you can't get fed, breathe, relax, and farm instead! :D

So I hope you have a bit more insight on how to build and/or play Jax a little better from reading this guide. I'm trying to help where I can. Anyway, enjoy playing Jax while you can; if the nerf bat continues to smack him he may just enter the Evelynn zone soon (Fingers crossed! D:)