Jax is a great champion, they're tanky, win most 1v1's and have AP and AD. They can stun and do a lot of damage, and you can often escape many fights. They hardcore counter AD such as Tryn, Yi and Xin late game. If you get fed a lot, you can easily 2v1 squishys.

I personally use flash and exhaust in twisted treeline. Flash is obvious, for getting away. Exhaust is for slowing them for ganks, ignite can be replaced.

In summoners rift, I use teleport and flash. Ignite doesn't make as much as a difference as teleport does in my opinion.

If you're on a summoners rift map, buy 2 health potions at the start along with your boots. You can buy a cloth armour first if you wish, but I like the extra speed, which can save your *** if you may be getting ganked.

For twisted treeline, I tend to not bother with potions. It's such a small map so leaving your lane doesn't take as long, therefore it isn't much of an issue. I also rush to the bottom bush for a gank, so I want to get there ASAP.

Get a Ninja Tabi because your stun requires dodge to work, and dodge can save you on many occasions.

Next I get a wriggles lantern. It has a ward in-built which you can place around and not having to worry about buying wards, it offers great lifesteal (18%). It also offers defence and some attack damage, and an ability which gives you the chance to hit an extra 500 20% of the time. It's overall a great item, thus get it early.

Next get a hextech, it offers lifesteal, and since you're a hybrid you need items that offer both AD and AP, so the hextech is good. It can also deal 300 magic damage and slow them for 50% for 3 seconds, which is easily enough to down someone in a gank and leave them with no chance of getting away.

The Guinsoo's rageblade is a great hybrid item, it also has a special ability which makes you attack faster.

You'll have a massive amount of lifesteal now, so you'll probably be staying in the lane for a while. A banshees veil offers mana and life boost, so you can stay in lane even longer.

Atma's Impaler gives an armour boost and also helps with crits.

After this, your build is complete. I just buy elixers so I have something to spend my money on.

Pros:
-Great against dps, can almost 1v1 anyone
-Can often easily get away, you can stun and run and also leap to minions
-Empower then leaping used as harassment can deal a lot of damage
-If you get fed you completely dominate and if you vs squishys you can easily 2v1
-Great for farming mid+ game
-Can solo dragon at level 7 after you get wriggles lantern and an amp tome

Cons:
-Runs out of mana a lot early game
-If your vsing a ranger that knows what they're doing, they can harass you a lot
-Early game isn't so good

Early game, just try and focus on last hitting minions. If you attack the entire time, minions might KS you. If you last hit, this isn't an issue. Throw in some empower leaps to harass enemy champions, but don't think you're invincible with it.

Later game, just attack the minions from whatever health they're on, since you do more damage you don't have to worry about other minions KSing you.

If your opponent has gone back to base, you want to try and hurt their turret as much as you can while they can't turret hug, right? Feel free to use counter strike on minions to push a little bit faster.

Let the tankiest person go in first, if it is you then so be it. Ulti, empower then leap on your target, and constantly normal attack them. If they attempt to flee, stun them or use your hextech ability on them, or exhaust them. If you're fleeing yourself, try and stun and flash if needed. Some people run after you when they don't realise you can just turn around, empower leap on them, and kill them even though they are the ones trying to kill you.

Look, if you're in a lane against turret huggers, what can you do? Not much. If you attack them, the turret owns you and if you attack the turret, they attack you. You can try and kill minions but the enemy champions just attack you and know you won't attack back because there is a turret there.

So what can you do? Lane swapping to a lane that needs help is an option, but other than that jungling can be great. The passive from your ulti, epic lifesteal, 20% chance of doing extra 500 damage, what more do you want? I admit early game jungling is poor, but late game can dominate. You can also counter strike and hit multiples of them.

Here's some tips when using Jax.
-When you place your lantern, you can leap to it along with other minions
-Use empower before you leap for extra damage
-Feel free to counter strike minions if there's groups of them
-If you have an enemy champion chasing you and you fear of dying, stun them if they're behind you
-Take use of your free ward
-Harass as much as you can with your empower-leap, it does a lot of damage
-You'll often win 1v1's, you should have epic dodge chance, so don't hesitate to stay and fight