Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
I call this the Diva Jax build because it is enormously expensive.
I'm not glued to the runes but I feel like magic resist is a really good compliment to Jax's active ultimate skill and keeps him a little more survivable against AP. His dodge will be so high that and AD may as well just run away. His dodge runes are a must and his crit chance, I'm not glued to either. I haven't checked if there is a cap on dodge and crit chance but if there is, I'm sure Jax will have reached it with his items and the crit chance runes can go to magic resist.
The hextecks are awesome because Jax is a hybrid AP/AD and so are the gun blades. A few hits on minions will send jax back to full health again and ready to tank another turret. Another benefit of these are that they have an active ability with them causing damage and preventing the target from dodging. I don't use it much because I tend to forget but I feel it can be the last chance button if you're ganked and only can take out two or three of the enemy champions.
Oh, also with the dodge capping. If this is the case, get the ninja tabi still but when end game comes near, get mercury treads(if those have magic resist) against a high AP enemy team or if its pretty balanced and you feel like you have enough survivability, get either swiftness or mobility. Heck, if you feel you're fast enough with two phantom dancers, you should just get Zhonya's ring for it's 2 second invulnerability. Good idea, I'll add that to the item build.
For the skill sequence, I'm not entirely sure if that's correct as I didn't write it down. Basically, though, you want leapstrike first and the dodge one. This will keep you pretty tough in the beginning.
As for game play strategy, I try to encourage my team mates to avoid group battles until late in the game that way we can farm a ton. Also, I try to get into a lane against other AD right around mid game. It's best if you can solo your lane that way you get lots of loot for the items.
After getting gunshoo, you are very capable of getting dragon so get it as much as possible- it puts more money in your pockets.
By level 18(sometimes before that) if you have 1 hexteck and 1 phantom dancer, you can kill the baron while your team mates are keeping the other team occupied. After that, take out a small group of neutrals to get your life back and get back with your lane to destroy the other team.
Lastly, you should be pretty vocal with your team mates telling them what you're up to and stuff that way they don't get mad if you're soloing a lane or something. You also need to keep sure that the other lanes aren't in trouble though