after playing some games with Jax and buying one of those: longsword, boots, or cloth armor + potions i started to think over these items, and a good replacement. dorans blade gives 6 damage which is almost as much as longsword, 120hp which benefits you more than the cloth armor, and besides some lifeleech which is nice too. well yea, it delays your ninja tabi a bit, but it benefits you in the level 1-4 stage because of the additional hp, while lasthitting slightly better -> catching up in gold.

Early Game:
Your early game is depends on their and your picks, if they got 1-2 melee you should lasthit more often, and stay at the line of your creeps.
if you are laning with a snaring hero, you can try a go at level 2 or 4.

when laning against 1-2 nuker/ranged you should be more defensive, leech xp and only try a lasthit if its safe for you to get away again (leap to your mate, ...) use leap to escape their engages and play passive.

Mid Game:
you should have your rageblade by now. - "throwing in a good advice": when you have 300 spare money buy an elexir of fortitude! - You should try to get the blue and/or red buff and engage on your lane or gank in groups of 3+. while you swap lanes or prepare for gank,you should try to get off some forest creeps if you can still stay in the ganking schedule.
Try to always farm gold somewhere, but only if its safe to escape or if it doesnt delay the gank (you could miss the right moment to engage).

when not ganking, try to get on abandoned lanes and farm creepwaves and try to push the tower. (again only if its save/ you can see at least 3-4 enemies somewhere else on the map)
try to intercept incoming enemy creeps right before the enemy tower range, so you can maintain your attackspeed when changing from creeps to tower by the time your own creeps arrive.

usewhen soloing vs a hero, so you can catch up and get off your 3rd - 6th hit on them.
or use it when running away. (e.g. use cutlas -> blink away + run; or stun + cutlas + run)

Late Game:your corebuild should be finished now and look like the above.

you should be getting in teamfights now very often. remember: all your spells deal magic damage which is why spellvamp is a very good ability on jax. you can heal yourself greatly with your splash/AoE spells, or every third hit with your ultimate.

Late Game - Teamfights:
in teamfights try to jump in with leap strike and empower, after your tank/initiator went in, or got focused. immediately drop exhaust on the enemy team carry because he will want to kill you as fast as possible.
don't forgetbut try focus off the stunner/low armor champions, because you will get focused and you will get stunned, so you need to finish them off quickly and clean.
after the teamfight when they are less than your team, if you have enough hp left, go to another lane and finish another tower off on your own. (if you die your team pushes in, if your team gets warded off, you should be able to push the tower)
with more then half hp you can also try to tank and destroy a tower solo (first intercept creeps out of tower range and gain attack speed, then switch to tower and destoy it)

Luxury items & examples:and evenare good luxury items depending on opposing team and game time.

if you are doing good in teamfights and dont get focused that often you should:
get a soulstealer, a bloodthirster, a void staff or an abyssal scepter so you can kill off enemies faster. i reccomend bloodthirster and soulstealer first
(on magic restis heavy team i HIGHLY recommend abyssal scepter and voidstaff, otherwise you will lack in damage on those)

when getting focused(nukes/stuns): get banshees veil, or a 2nd phantom dancer depending on the damage type you are receiving.
normally its phantom dancer vs nukes (physical and magical)
and banshees veil vs a stunheavy team.

when fighting dodging enemies: get sword of divine

when fighting heavy tanks get mandreds bloodrazor and later on sword of divine for dealing a significant amount of damage, and not missing them. (guaranted %health damage amount with every hit, because they cant dodge)



REMEMBER:
all your skills scale with AP, and are magic damage (that + the 50% slow is why i chose hextech over bloodthirster) so the 13% spellvamp Will heal you
you benefit from magic penetration (void staff)
or the aura of abyssal scepter.

Conclusion:
Due to Jax gaining magic resist with each dodge chance, i recommend for most of the games, to get a bloodthirster and a second phantom dancer, so you have a better damage output while having survivability vs magical nuke champions or physical carries.

have fun playing Hextech Jax!

P.S.:
!please add feedback, as im still trying different luxury item combinations!