I'm no pro but this is how I play Jax. I like to jungle him, but you will need a decent leash at blue if that's where you start. If you don't jungle then I'd suggest swapping out Smite for Flash.

Hopefully it's obvious as to why I chose these runes. If it's not then study up further about Jax.

Putting 21 points into a mastery tree is silliness. Putting points anywhere put defense/utility is silliness unless you're jungling.

Don't follow my item build to the T. Buy what you need or lack at that moment. Getting a wriggles and selling it late-game is beneficial for jungling but you gotta be sure you're getting gold income. This is not the order I buy items, I usually try to get a blasting wand before I get a giant's belt, then I like to finish rageblade. But it just depends on how well I'm doing.

Q and W are your main skills with 1 point in E at level 1. Try to get Q and W up ASAP while never neglecting R when she wants some attention.