sorry if it's a little dirty, im new to all of this.
constructive criticism is much appreciated.
basically this is my hybrid jax build, and it has received much testing.

this set-up is normally for a solo jax on a 3v3, but works well in 5s as well.
IF you're solo jax on 3s, the build varies a little though. instead of going right for ninja tabi, i'll buy a vampiric scepter. this gives me the ability to stay in the lane until i have enough to purchase the rest of my ninja tabi.
solo jax dominates the jungle in 3s, so use that to your benefit. red should be grabbed at level nine, and dragon grabbed after grabbing red. obviously, use discretion when you're jungling for red and dragon. getting dragon last hit by an enemy player is an unfortunate consequence and one of many if you don't.
for 5s, i recommend laning bot with this build. the dps this build puts out makes controlling dragon fairly easy. grab buffs and dragon often, but make sure not to ignore your lane in doing so.

i couldn't be bothered to try and put an exact build order for him, so i put the items there. i always get ninja tabi first, followed by hextech gunblade. from there the order can vary based on how the game is going. manamune also isn't crucial to the build, so if you want you can switch for a defensive item. randuin's omen, thornmail(depending), and guardian angel are all good defesive items for this build.
also, my skill order might be a little wonky, but w/e. i try to aim for balance throughout leveling.
with my rune build i'm looking to up my dodge chance, while giving me some CR and AS for increased damage output.
for masteries, i go mainly defensive, this is to make him able to deal with damage a little better as i don't normally build defense items. the rest is focused for magic pen, so that abilities tickle a little more.

due to the dps output this build can have, you'll often be prioritized late game. in order to prevent from being squished in teamfights, try to hold back and wait to engage until a clear target has been set, then jump into the action and use your stun to control how each subsequent target is engaged. don't be afraid to get in 1 or 2 v 1s though, especially if your heal and ignite are up. you can drop them to finish the first target and the hextech on the second target. i mean obviously it depends on the target/s, but you get the idea.

it's pretty simple, just auto attack and get last hits. late game counter strike becomes an amazing farming tool though. just stand in a big group of minions with a dodge up and watch them drop.
jungling is also pretty simple for this build once you have lifesteal.
jax is an amazing buff-*****, so don't be afraid to exploit your jungle if it's open.