Here are my ranked stats with Jax for Season 2. That means only Solo Queue and Ranked Teams. The games may seem few but there are lots of customs (for tournaments) and smurf games not taken into account.
Jax’s early game is mostly about farming. You'd rather get to level 5 or 6 before you start trading with your lane opponent. Meanwhile you can stay safe from ganks using your E and Q.
After you get your first item or two, you can probably 1v1 anyone in your lane, provided they haven't gained an early advantage on you. Don't play risky trying to get kills, eventually you will out scale everyone. Keep in mind that every time you use your Q/E offensively, you are vulnerable to a gank, so try to have a ward up at all times.
On team fights, use your E to engage offensively and try to get a carry stunned. Use your ultimate only if the enemy team starts focusing on you. Your aim is to stay alive as long as possible and deal the most damage you can. Don't hesitate getting out of the fight for a bit to re-engage with your Q once the focus has shifted.
Finally, if you are able to, try to lock the enemy AD Carry down and kill them. You can stick on them quite easily with Trinity/Βlade.
Jax is a simple hero with great capabilities. The hardest thing about playing him is deciding when to go in and when to come back out. You will get used to it after you play a few games with him.
Tips & Tricks
Using W resets your attack animation so you should try and use it right after you attack.
While in the laning phase, after you get your ultimate, the most effective way to harass your opponent is:
Get two hits on minions thus gaining two stacks for your ulti's passive and then jump on the opponent. After you land, hit W. This will proc the ultimate's damage as well as your W resulting in large amounts of damage.
Q can be used to jump on friendly creeps, champions and wards when you need to escape. So always try to carry a ward or two with you.
Jax is really good with Baron and Dragon. You can sneak an early Baron with just one or two teammates using his amazing sustained damage, dodge and ultimate.
In teamfights, if you can jump on the enemy AD Carry with counterstrike active and get the stun in, you can probably kill them or chase them away from the fight.
Jax does amazingly well against Auto Attackers.
Jax has many synergies and not a lot of counters.
He can work very well with Morgana due to her black shield. Also, her ultimate allows him to attack people while they can't turn around and focus him as they are risking a stun.
Zilean is also quite good as the movement speed buffs allow for better mobility and provides a free Guardian Angel to boot.
Lulu is very strong with Jax as she can chain her ultimate with Jax's E for tons of CC while also protecting Jax allowing him to attack more.
Finally Orianna works quite like Lulu. She shields Jax as he jumps in and then uses R for deadly Ultimates.
On the counter department, Jax has troubles with only a few heroes.
Yorick: Top lane's bane -recently fallen out of favor- can almost completely shut down Jax. You have to buy sustain items immediately. Vampiric Scepter, Doran's Shield, Mercury's Treads. Then just focus on farming; you can easily kill Yorick when your jungler ganks.
Darius: Another hard opponent, try to farm passively but when he pulls you/uses his cooldowns, you must fight back! Get some damage in with the stun and then fall back, his cooldowns are shorter than yours. Go for a Doran's Shield, Ninja Tabi and Vampiric Scepter early.
Vladimir: Quite annoying as he can harass you a lot and has also a lot of sustain in lane. You must manage your resources carefully and not hesitate to recall early for items. Get a couple of Doran's Blades, Mercury's and maybe a Hexdrinker and go put pressure on him.
Malphite: People believe Malphite is a hard counter to Jax. That does not apply. Swap your quints for health regen (This is a must). Rush Mercs and Sheen. Harass him with W or Q-W and dont stay in to fight. Just hit him with magic damage and back out. This way you should have an even (or even better) match-up.
Special Mention: Nunu
Nunu goes into both categories. His buff is incredible on Jax, making him a killing machine. Unfortunately, when Nunu is on the opposing team his E makes Jax really sad.
This applies for both jungle and support Nunu.