My build for Jax. Probably not what you would build because it's weird huh? It works great though! Sorry I don't have any visuals, so I'll keep this short and simple for you guys to read through.

Pros:
-passive grants great survivability even without tanky items.
-deals massive damage due to ultimate.
-tri-force grants movement speed, allowing Jax to usually catch up, leap-strike, and kill.
-solo's baron with only 3 items, usually before an entire team even goes for baron.
-fun as f getting kill after kill.
Cons:
-starts off slow at farming.
-doesn't deal great damage until ultimate is unlocked.

Seals: dodge; obviously for E skill and survivability
Glyphs: cooldown (per lvl); very useful against escaping noobs
Marks: armor pent; helps deal great damage (especially) late game
Quints: flat health; for that head start early game. you can substitute for any other you think would be good

21 / 9 / 0

You want that offense, and some decent survivability as well.

This is just a guide to help you; final builds should depend on composition of the opposing team.

That being said, with only 475 gold, I start off with a Doran's shield and a health pot just because it helps the most compared to items that can be built into something else.

I often find myself substituting Banshee's Veil with either Guardian Angel, or another Hextech Gunblade. This depends on how much survivability I need. I tend go go towards Banshee's Veil against nukes, and Guardian Angel against high crowd control. I make another Hextech Gunblade if I have no trouble with dying and/or they have a few survival-tough champions. It really comes down to what you think.

NOTE: The items statistics provided by the website are currently not up to date with the time I made this guide. Therefore, this affects whole stats listed above. (EX: Hextech Gunblade has been buffed to 60AD and 75AP)

Start off with W to help your farm up, next Q to harass, and third E to complete your combo. Max out W first so you can farm quicker, as well as deal a good amount of damage. Next, Max Q for additional damage since you use this skill to catch enemy champs anyways. Finally, max E because all you really needed was 1 point on it for the stun. Take R whenever possible.

Remember Jax's passive will actually give you more health in addition to the flat amount listed above.

Exhaust because when you're up against DPS, your stun just might not be enough. Just be sure not to use it too early.

Flash because you just need to evade sometimes, and there's not always something to leap to.

HEADS UP: Using this build, Jax can SOLO Baron with only Ninja Tabi, Hextech Gunblade, and Trinity Force! Try it :D Just have the team on standby, pushing top and middle lane in case it's warded. If they catch you, you can always run into Baron's camp and flash through the back of the wall, or my personal favorite, you can drop a ward behind Baron's camp before you start attacking it (this also keeps an eye on scouters), and Leap Strike to it over the wall to escape, also leaving flash for an additional juke if needed. [:

Jax is great with picking off players, but that doesn't mean jump into a team fight first. Jax isn't a tank, so let the tank initiate before you pentakill. :D

Let me know how the build works for you! I would appreciate comments on the build, thanks.