Well I know you guys don't like reading intros so I'm just gonna say, I'm a Jax player who likes playing jax.

I'll be honest, many guides tell you to max W first, I don't, this is because Jax's Q always has better scaling and damage then W in about 70% of short exchanges. Plus it gives you more mobility then leveling W. Say you get ganked, Jungle was probably waiting for you to use the Q as he thinks it is on a 10 sec CD. HOWEVER when maxed its only on 6 sec, so as long as you put your E up, stun, then you should always be able to get away from the ganks, without flash. Also most of the time you only get to hit W ONCE in an exchange. I admit in theory you will get more damage out of maxed W then maxed Q in 10 seconds, HOWEVER in short exchanges, maxing Q will deal about 50% more damage then maxing W, the biggest difference other then damage would be mana cost, Q needs 65 while W needs 30 and they stay the same in all levels. This short exchanges are key to winning your lane, look at reneckton, right now he is the strongest top laner, this is because despite his high CD on his spells he has high burst and sustain for short exchanges.

IMPORTANT: WAIT BY THE SHOP (VERY EDGE OF IT) AND BUY AN EXTRA POT BEFORE YOU GO TO LANE, IF YOU EXECUTE THIS GOOD ENOUGH YOU CAN GIVE A SMITELESS LEASH AND NOT MISS OUT ON EXP

Now this is where it gets really fun, usually you would want two Doran's blade for early health, damage and sustain.

Then you rush TRINITY FORCE, you don't even need boots, jax has base MS of 350, with phage you should not need boots.

After you get a Trinity Force, go buy Bilgewater Cutlass. IF you are short on Gold, sell your Two doran's blade and your flask.

HOWEVER if you have about 400 gold after Bilgewater buy a boot, It will give you 384 MS, which is more then enough with your 6 sec gap closer.

DELAY BUYING BOOTS AS LATE AS POSSIBLE.

When buying boots consider their team's cc and damage type, When there is more then 3 hard cc buy merc treads, if they have 2 or less hard cc with rest cc being slows, get boots of swiftness, If their team has lot of AA (Auto Attack) damage champions (Master Yi, Aatrox..etc), get Ninja Tabs, at late, LATE game you can get either CD boots or an another item, your call.

Something that CAN be great or BAD for Jax depending on the match up

Before 6

Harass only a little bit, unless you are pretty far ahead. When they come to AA poke you, hit him with your own AA then W combo then put your E up and chase them hitting them with your AA and W, you should be able to stun your opponent and get one or two free AA off. But usually don't try to all in unless their mana or hp is low. ALSO CALL YOUR JUNGLER FOR A GANK OR TWO, early kills on Jax means, easy laning phase and an easy game in general.

After 6

This is after you get your ult. This is when the real fun begins, this is when you have your phage or sheen or both! So what you want to do is to AA creeps twice, Q or walk over to their champion get the ult proc off and use W to AA cancel, dealing tons of burst damage (This is the hit hit doush). From this point on you should be able to win most of your trades.

OK I'LL BE HONEST DON'T GO IN BALLS DEEP IF YOU DON'T HAVE FULL BUILD BY 20~30 MIN. But just wait it out by staying beside your Mid or marksmen and protect them while waiting for opponent's carry to be low (Below 50%) or when they are out of position. So basically just clean up.

HOWEVER YOU CAN GO BALLS DEEP TO THEIR FACE after about 35 min when you have your Randuin and Mars. You jump to their carries, hit them, stun them, chase them with everything you got and kill them or die trying, HOWEVER by this time your team should have been able to break through enemy's front line and able to clean up the rest of the opponents.

But usually go in little after their hard cc has been used for better results.

Singed is actually not a champion that can HARD counter Jax if you play a little passively. Usually you want to Turtle until 6 and use the hit hit DOUSH combo to widdle down his HP, when he has less then 150 mana its your go to kill. Before 6 when you are lv 3 is one of the strongest point Jax is against Singed , Harass him with your full combo but don't start with Q, go up to him, AA, W and put your E up if he hits you back, he will fling you, this is good, your saved your Q for this, Q back to him for stun then AA. This should give him more damage then the one you receive, do this until lv 4 and gain advantage over your lane. In this lane you might want to get a bilgewater before getting your trinity force because of poison damage and to chase singed better.

Gangplank, I personally think Gangplank is an easy lane, he pushes lane naturally so just farm under your turret until 3, remember your E blocks his bullets and that it is his only strong damage source. Always stay above 50% hp, after 6 you can kill him like there is no tomorrow.

Reneckton, he seriously wrecks Jax's game, he has more damage and has a better laning phase them him. HOWEVER you can still beat him by having your Jungle camp top and only engaging if his q or w is on CD, then you can out trade him by alot. Other then that, don't be cocky and turtle, you can never fight reneckton until your have your Trinity and Blade of the ruined king. Even then you can barely kill reneckton as in you kill him and he kills you.

Ok so if you fight your opponent early and get them below 50% hp, and you do that constantly (Meaning keep the pressure with short exchanges), you basically win that lane because you have so much sustain from flask and potions.