Hello, I have written this guide specifically for 3v3. Unless your team comp is bad, or if your teammates are utterly clueless, you shouldn't go wrong with this guide :) If you want to try this build in 5v5, feel free to let me know how it works. In theory, it won't work as well but you never know!
ALSO, please remember that this game is all about strategy, there isn't a set 101 build obviously. Take everything w/ a grain of salt, and use your own head to think in situations. Okay? Okay. Have fun with this build, enjoy!

Pros
You're OP Jax
You eat away at the hp of squishy champions.
You're OP Jax
You're an offensive hybrid, can easily build your items to counter the enemy's item. E.g. if they have thornmail, build more AP.
You're suchhhhhh an OP Jax

Cons
No range to harass with.
Early game may be a struggle.
Takes a good amount of time to farm your core, so it's extremely important to solo and/or get fed.

Even though I'm sure people will rage and Q.Q over this, let me go ahead and say that runes aren't as helpful as masteries. Of course they make a difference, but they're not the end all be all of your game. BUT, since these little stones exist, I'll go ahead and explain my setup.
Greater Mark of Desolation: Apart from your initiating Q and every third auto attack from your R, you're gonna be auto attacking in fights, right? So arpen is a little more important than magpen in my opinion. Go ahead and take these, you won't regret it.
Greater Seal of Evasion: Haha, out of all the champions, dodge is most important for Jax hands down. Procs beautifully with your E.
Greater Glyph of Focus: A little extra cooldown never hurt anyone. Having your Q and W accessible is always handy, especially when you're relying on it to chase down scared enemies. YOU CAN ALSO TAKE GREATER GLYPH OF POTENCY. There, I said it. No reason to rage now :) Or you could mix Potency and Focus together, whatever you want. But whatever Glyphs you take, please don't take anything else.
Greater Quintessence of Desolation: With Jax, you can really take any Quintessence that procs with your Marks, Seals, or Glyphs. I just always had Desolation, but Insight or Potency work wonders as well. I wouldn't advise wasting the IP on Evasion Quints unless you really want, since you already have enough dodge chance. And since there's a cooldown that deprives you from spamming E, it takes away the need to have that much evasion.

For masteries, I take the standard 9/21/0 defense setup. Pretty self explanatory.

Q: Why not 21/9/0?
A: This offensive setup can work, but it's just not my style. In early laning stage, I find it helpful to have 21 in defense and outlane that annoying opponent. Since my Jax is hybrid, I don't find it completely viable to get anything past Archaic Knowledge. In mid and late game, your Rageblade plus Hextech Gunblade should be more than enough to crush opponents.

Almost always take Ghost. Put it this way: if everyone has Ghost except for your slow butt, that's gg for you and it's a guarantee kill for the enemy whenever they have their Ghost up. While your other two teammates are running away with their Ghost, you'll be eating their dust. Not good man! Of course, this is in theory. And of course there's exceptions. If you're using a champion like Singed and you want people to chase you so they eat large doses your poison, take another evasive spell like Flash. But for Jax, you don't have poison or anything of the sort. You're a head to head fighter, you go in there and take them all out because you're OP Jax. So when you make a bad maneuver and need to get out of there, Q away from the fight and Ghost your butt out of there. As for being offensive with Ghost...I would usually only use Ghost for chasing if the little bunny you're chasing is Ghosting too. Once again, between your Gunblade's active, Exhaust, and Q, there are only a small handful of champions that can escape you and your lantern. So save Ghost. Unless you're feeling unkillable. :)

Exhaust: It just works w/ my playstyle, very handy spell to have. You can't really go wrong with it. And since you're worrying only about 3 enemies and not 5, it's even more viable in 3v3 for getting away or chasing and completing a kill.

Other good spells:
Flash: Always good in 3v3. Make sure you adjust your playstyle to fit it, so you'll have the enemy crying "flash is such a noob spell". This is pretty easy to do in 3v3. Almost always stand next to a wall, no matter what. If you need to get away, just hop over and problem solved. You can also use it offensively to cut off enemies or chase them down when they Flash over a wall.
Heal: Yeah yeah, it's a noob spell. Don't jump all over this just yet. If you're new to Jax or not completely confident in your skills with him, taking Heal can be a big plus for the early laning stages when you're soloing. Who knows, it might even get you first blood. :)
Ignite: If your other two teammates have Exhaust or something else and you feel like Ignite might help, feel free to switch this spell for Exhaust.

The 'funnest' part to Jax :D
Chances are, you'll be soloing in top lane. I go with cloth armor, ward, and one or two hp pots depending on how handy I feel like they will be. Place your ward in the center at red buff when the time hits 1:30. Yeah, it sucks that you need to burn 75 gold on this item, but chances are your teammates won't think of getting a ward :( BUT, once they see that you're warding, they might help buy wards after you ask. As for cloth armor, it gives you some defense to let you stay in lane longer and it builds into Ninja Tabi too. Double plus! Also, if you're laning against another Jax, chances are he'll buy boots or Doran's Shield to start off. If he bought Doran's Shield, that just sucks for him cause he won't have health pots like you do. So go ahead and harass his health down. If he bought boots and some hp pots, that sucks for him too cause he won't have that extra armor on him. And trust me, it makes a big difference.
On your first trip back, grab a Blasting Wand. I prefer the Wand over the Pickaxe because it procs with your Q, which is relied on more when you're trying to harass and burst the enemy down in the early laning stage. If you're a boss and somehow got enough money, get your Ninja Tabi as well. If you didn't get that rich yet, no sweat, just get Ninja Tabi on your second trip back. I usually don't like wasting a trip to get Boots of Speed before upgrading to Tabi. You already got Cloth Armor, so it shouldn't be too hard to get Tabi in one shot after that, you know?
After completing Tabi, finish your Rageblade. Then work your way to a Hextech Revolver (it's cheaper than Bilgewater Cutlass). If you have extra money left over from the trip where you completed Rageblade, and your team or enemies are ganking a lot, grab an Amplifying Tome w/ that extra money. From Hextech Revolver I usually try to complete a Gunblade. But if you really need a Bilgewater Cutlass for whatever reason, you can go ahead and get that first. Make sure to activate the Cutlass/Gunblade! I would use that before using Exhaust since the cooldown is significantly shorter and it deals 300 magic damage too. Good stuff :)
Your core build is comprised of Ninja Tabi, Rageblade, and Hextech Gunblade. At this point, you're probably either dominating and making the enemy complain about how OP your Jax is, or they're winning because your teammates fed them on the bottom lane. Here are some viable options for your next item.
Rylai's Crystal Scepter: Everything about this item is great for Jax. After your core build, it shouldn't be hard to farm this item up either.
Guardian's Angel: This item turns some major tables in late game, especially in 3v3. You should be able to take out all 3 enemies at once if you got this item. And if you can't, your teammates will hopefully be there to help you finish up the job.
Phantom Dancer: If you want to finish the game and just need some extra attack speed for turrets/buildings, hit this item up.
Thornmail: If the enemy team has a lot of AD or auto attacking champions like Master Yi or Tryndamere, get this item and let them kill themselves.
Banshee's Veil: Really good for the sole purpose that it will cancel out the enemy's first ability so you have a chance to get your Q on them. I'd suggest getting this if the enemy team is magic heavy.
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Items that would work but I don't recommend:
Trinity Force: If you feel like the game is still going to take a while, you could get this if you realllly want. PLEASE NOTE, I usually hate getting Trinity Force in 3v3 no matter what champion it is. For 5v5 it's a wonderful item, but the time that it takes to build in 3v3 just doesn't make it worth it.
Lich Bane: Yeah it's a great item for hybrid champions, but you're not getting your full money's worth on this one in 3v3. You shouldn't be having mana problems, and the map is small enough that you can afford to tele back to base if you really need to. Also, you're not spamming abilities that much either. After my core build and possibly Rylai's, I usually go defensive. It's more viable - Jax already has enough damage output :p For just 400 more gold or so, you could be getting Guardian Angel rather than Lich Bane.

Take top lane, unless you have a stubborn teammate who insists on taking it. At 1:30, make sure you ward the red buff. Do NOT waste your mana on Q until level 2 when you can proc it with W first. Jax deals so much damage this way that you should have no problem against most of the squishy carries that will take top. After you get E at level 4, check their items, the bottom lane (in case the enemy tries to gank), and summoner spells (especially for the annoying Heal); when you harassed them down, feel free to go in for a kill. Or you can fake it and pretend that you're about to try to kill, but back out and see if they waste a spell, hp pot, or a defensive spell.
Jax gets out ranged by many champions who will take top, but you just need to confuse them and control the minions. For example, if you're laning against someone with a shield spell like Sivir or Nocturne, get close to them and make it obvious that you just hit your W. They'll cast their shield, BUT your W's effects will outlast their little shield. So stay in range of them, and as soon as their shield wears off, Q them. Because they probably got scared and began retreating when their shield wore off, you just successfully zoned them out of exp range too.
If you extend too far, go and jungle while they push the minions back. Don't waste time jungling for buffs though, just get some easy exp and gold from the neutral monster camps on the very left and right in the top jungle as well as bottom.

Within 25 minutes or so of the game, you should have completed your three core items: Ninja Tabi, Rageblade, and Hextech Gunblade. You want to spend as much time farming as you can. If you're not farming, hopefully you're ganking. Basically, farm the minion waves. Once they're pushed, go jungle or go to the other wave. Make sure you ward or have a guess as to where the enemies are if you do not have visual on them. I've seen way too many junglers die because they refuse to buy a cheap ward or try to solo the dragon. I actually don't focus on the dragon that much, I focus more on outleveling the enemies rather than on buffs. In teamfights, you want to initiate and focus the squishy one. If you get caught in the middle of the fight, hopefully their attacks will proc your E and you can Q away if you're low on hp. Don't run all the way back though - stay in the area but out of the enemy's reach, because if your teammates manage to get the enemies low on health, you can burst them in one shot with your W and Q combo. If minions or monsters have made your E available, you can also jump into a teamfight with W-->Q then instantly hitting E to stun all the enemies.

Staying with your teammates to try and push the enemy turrets makes you a really big threat. If you're confident you can fight off enemies by yourself, go ahead and push the other lane. I would advise to take out BOTH inhibitor turrets before destroying the inhibitors. If you take out the inhibitors and don't finish the game soon, they'll just feed off of the minions. With your core plus Guardian's Angel, you can be extremely offensive (but don't be reckless).
Also, CONSTANTLY check the items that everyone has built. Late game is extremely important so you want to make sure you know your chances of surviving in a fight. I've seen people go from having no enemy turrets destroyed to winning a teamfight in late game then tower diving all the way to their Nexus. That is why Guardian Angel is such a major item. Imagine if your whole team has it, the enemy essentially needs to kill six people instead of three.
No matter how tempting it is, do not jungle or wander off by yourself if you don't know where the enemy is lurking. It's not worth killing those few extra neutral monsters if the team gets a kill off of you.
But since you are OP Jax, if you're confident that you're unkillable and your kill/death/assist ratio is backing you up, you can be more aggressive. I've won a 1v3 against the entire enemy team with just my core build because I spent a lot of time farming and leveling up.

Jax isn't a complicated champion to use, and there aren't a lot of different things you can do with him like Shaco, who is much more complex. But just like all champions, there are some tips that separate the pro Jax players and the inexperienced kids who just want to play an easy champion for an easy win.
Your standard combo is...well, not much of a combo at all haha. Initiate with W then Q, hitting E ASAP. If you're dealing with an AP enemy, W and Q then R as right away.
If you're in a 3v3 fight and you're winning (because you're OP Jax), save your W and Q when you feel like the enemy might start making a run for it. Use it to get in range of them and finish their sorry butts off.
Remember that you can use Q on enemies AND friendlies. This is where the pro Jax moves come in. Actually, it's also probably the only reason you'll need to rely on your teammates as well =p Q to your teammates or minions when you need a quick getaway, OR when you're chasing an enemy down and your teammate or minions are in between you and your target. Remember that walls mean nothing to you, because you're OP Jax. In the early laning stages when you and the enemies are making ganks from the bottom lane to the top lane, tell your teammates to stand next to the wall so you can Q over it. You can create ganks before your enemies even realize where you are or how it happened. If you have visual (a ward or teammate) on the dragon, you can use Q to jump that wall as well. Stay next to the wall when attacking the dragon - if the enemies come and try to fight, you can escape by targeting a minion with Q to jump the wall again. KEEP IN MIND THAT IT MIGHT NOT ALWAYS WORK - the minions may simply just be out of reach. Common sense - please check minion placement before standing next to the wall when getting dragon. Cause if you can't jump the wall or Q to a teammate near dragon, you're easy bait.
Another tip that's really helpful but not always possible is to have your R and Rageblade stacks up before going into a teamfight. If you know you're about to get in a fight and you're near the minions, hit some of them to get your procs up before jumping into battle with your Q.

Remember, use this guide but adjust it to your playstyle. If there was really a perfect guide for each champion, how messed up and boring would that be? The game is all situational, but I hope you can get some good ideas/strategies/etc. from this :)
Personally when I learn a new champion, I'll look at different top rated guides on different sites then combine my favorite parts about each one of them to suit my playstyle.

Oh, and for those crybabies and trolls that are lurking around League who complain about Jax being OP: Jax is very vulnerable before his core is complete. One reason people think he's so good at 1v1 is because of just how wonderful Rageblade and Gunblade proc on him. It's almost like Riot designed those items specifically for Jax. And it's not an easy core to rush or complete either. It's their fault they let you farm all the way up to get to your OP stage :p