The first build is for attack speed, killing quickly. The second is building him for defensive to live through more and do more damage each hit. You can choose either which one your closer to completing or which one you like more. Basically you can play him however you want, but this is how i do it. Get your ninja tabi quickly so you can move faster. Farm all over the map top and bottom for 3v3 or top mid and bot for 5v5. Dont take everything but take some minions here and there and level quickly, while giving every lane support in the process. When you get to level 6 start jungling IF you dont have a jungler. If you do, do NOT take there experience. But at about level 9 or 10 take dragon for your team if no one has yet. Also by level ten start trying to steal the blue and red buff on the enemys side. This is why i always get wriggles lantern. Use the ward it gives you and make sure you dont get ganked in the process of taking there buffs. If you do this successfully youve helped out your team a lot by now because you have the other teams buffs and you got them the gold for dragon. While helping support each lane. Do not stay stuck in one lane as jax UNLESS your lane is getting wrecked. Then you need to stay there, but thats really only if you have terrible teammates and what not. Well hope this guide isnt to terrible, it is my FIRST one so dont hate to much :P

for the first build i like to have all attack speed except for the glyphs. I like this because attack speed is key when trying to kill people really fast. Yes his passive and guinsoo make him faster over hits, but when you start fast you end up being even faster taking down people with ease throughout the game.
the second build i have strength for the marks and glyphs to give you some edge, but with the seals and glyphs being the same as before. I think you alwasy need a little speed and cooldown for most characters. Thats just my opinions though.

for the first build its just getting all the attack damage you can suck out of the masteries, pretty much all i can say. I mean you get every attack and life steal and cooldown from the attacking of masteries, as well as some extra turret damage which will help trying to win the game in the long run.
The second one is a little more defensive taking only the needed attack masteries, and putting the rest of them in defensive so it will help you sustain living a little bit more abd give you some extra health.

FIRST-

NINJA TABI-fast way to get some speed and defense

WRIGGLES LANTERN-good life steal and a free ward, very important for jax

GUNISOO-good unique passive to go with the passive jax already has + AD and AP

MADRIDS-armor, att damage, att speed, and 4% life with each hit, why not get it?
black cleaver- att damage, speed, and lowers armor, which is great

THORNMAIL-just amazing for any dps champs

SECOND-

MERC TREADS- great for anyone magic, if theres no magic people of course go with ninja tabi instead

WRIGGLES LANTERN-same

GUINSOO-same

TRINITY FORCE- well this just gives you a massive amount of stuff, which is great for jax consiidering hes attack and magic, plus health and just tons of good stuff

FROZEN MALLET-this is great to get a lot of extra HP for jax to sustain more in team fights, also slowing people is ALWAYS good for everyone. This will make it easier for you to get in a team fight earlier do damage and get out alive

THORMAIL-same

For the first build i wanted to get leap strike finished first to have it be on low cooldown to get to people fast and start destroying them. next is empower because its nice to have that doing a bunch of damage especially getting close to end game. Next is counter strike, yes it does some damage but early to mid game all you need that to do is stun them, so if your good at stunning people then your set. Finally the ultimate is always at 6-11-16 so that doesnt change.

For the second build i wanted to get empower finished first because this build is built more for damage and keeping your lane with alot of harass. After that is leap strike because this is needed to be on low cooldown at least by mid game, no later. The counter strike for the same reason as build one, all you need is the stun. Also same for the ultimant, 6-11-15 like always

for me i always use flash ghost and exhaust, unless i am a support character. The first one has two moving spells so you can obviously get to them quickly or run away quickly and either kill them or run away effectively. You should almost never die with ghost flash a stun and a jump. Same goes for the other set though. If you exhaust them then ghost it outta there hopefully with minions to jump to, you should be set!

as i did say before you do need to farm alot but ill try to explain a little better as to what i do in a game. I am always bot idc what anyone says, unless there my actual friends then i dont care but for this ill use bot. I got bot and always try and get first blood without overextending to much. If i get it i get i dont i dont. Anyways after i go back for full hp i kill at least 2 or 3 waves with my teammate in my lane. Depending on how many your killing try to get to level 6 fast. Then either wait to try and gank the other people on bot with mid, or go jungling if there isnt a jungler on your team take everything, do red then golems and camp. go back get full hp and do wolves and blue. Then you should have wriggles by now so go and get the other teams buffs without getting caught. This will either give you more buffs, or you can tell your team that they have a jungler. If you have a jungler yourself go and take the enemys first off, or just tell your team that the enemy has a jungler. at level 10 take dragon and then take it every 6 minutes from then out until mid-late game, extra gold means extra items! for your whole team. by level 17 or 18 being built correctly you can take baron easily. So go do that. While doing this you will either ward and get baron for your team, realize the other team already has it, or just get jumped 5v1 and get destroyed. Lets hope for the getting baron buff for your team haha ;)

Just follow this and pretty much makeup your own game plans during to help the most effectively you can as SUPER JAX! have fun stay safe and dont get to mad when you lose, because it was probably because of your teammates ;)