This is the build I use for playing Jax, I do jungle with this build and CAN get Baron at 20 minutes. You do not need a large amount of kills to obtain enough gold to be able to purchase this build.

The runes I use are...
Marks: Armor Pen x9 (1.66 Per 1)
Seals: Attack Speed x9 (0.76% Per 1)
Glyphs: CDR @ 18 x9 (0.9 @ 18 Per 1)
Quints: Armor Pen x3 (3.33 Per 1)

Total Armor Pen +25.
Total Attack Speed +6.8%
Total CDR @ 18 -8.10%

You do NOT have to use this rune set up, it's just the runes I have for my AD rune page.

EDIT: I changed it to have Dodge seals & Quints along with 1 movement speed quint. I find movement speed quints with Jax to be pretty nice for catching up to leap ect, and early game running around without boots.

3 / 27 / 0

Put 3 points in Offense, 1 in Smite and 2 in Archmage's Savvy.
Put 27 points in Defense, Max out: Resistance, Hardiness, Strength of Spirit, Evasion, Nimbleness, Defensive Mastery, Harden Skin, Veteran's Scars, Ardor, and Tenacity.

Start with a Cloth Armor and 2 Health Potions. If you have to recall before you can buy a Madred's get a Long Sword (Builds into Madred's). Get a Wriggles Lantern, Then Ninja Tabi. The next thing you get is Hextech Gunblade and then Bilgewater Cutlass. This is your CORE item build. For the last 2 items you can ether get more survivability or more damage dealing items, Frozen Mallet is an item I recommend just because there is basically no way that someone can escape you, you have your leap, your stun, and your Frozen Mallet which will make you able to chase them with ease. Force of Nature, Thornmail, Quick Silver Stash, and Banshee's are some good items you can get to have more survivability, you can also get a Randuin's Omen for the active and CDR.

Your skill order should go something like this...
E W E W Q R E W E E R W W Q Q R Q Q
EDIT: Max your Q before your W for the CDR on your leap!!

I typically try to max my E and Q first, maxing your Q after your E will bring its CD down so you can use it more often, try not to lead off with the leap that way if they flash away you can leap to them still, and your E will help with dodge change & stun for jungling and ganking.

I normally take Ghost and Smite.
Ghost: Good for chasing and escaping or being running back out to jungle faster I guess.
Teleport: Pretty much only good if you like trying to backdoor the enemy team, also good to counter people who try to backdoor your team though.
Flash: Pretty stupid IMO, your Leap is a free flash, and you can jump to wards. Just use your active on Wriggles lantern then leap to it.
Exhaust: You could use this, it would help with ganking I suppose, but keep in mind it doesn't blind the target anymore.
Ignite: You can use this to pick up some kills on low people... W -> Q -> E Ignite run off if low enough hp.
Smite: You kinda need it early game for jungling.
Clairvoyance: You can get this to check for ganks, but your support should be the person that has it.
Heal: You don't really need this...
Clarity: If you have mana issues this is a possible solution.. but IMO you should just use your mana more wisely.
Revive: Pretty awesome spell, most people frown upon it though.
Fortify: Leave this spell for your tanks.
Rally: It's an ok spell if you remember to use it in team fights... but it doesn't really help.
Cleanse: Pretty good summoner spell, but if the other team has a lot of CC you might as well get a Quick Silver Stash, the active has a lower cool-down than cleanse if I'm not mistaken and QSS removes Suppress unlike Cleanse.

Start with Cloth Armor and 2 Health Potions, get your E (Counter-Strike), go to Wraiths, smite the big wraith, kill the others, you should only have to stun 2 times. Go to the Wolves next, I attack the big wolf first, get your W and go to Golems, kill them and repeat. You should only have to use 1 health potion each time you go through the route at early game. Once you get Madred's you can get the "epic" monsters with ease, jungle until you can get a wriggle's Lantern. Get your Wriggle's Lantern and get the buffs again, if your on top half of the map get Red then Blue then Dragon, if your on bottom half of the map get Blue and then Red. If you have a good amount of health left you can try to gank or hold a lane for someone. Next item you should buy is Ninja Tabi. Once you have those you buy a Hextech Gunblade and then a Bilgewater Cutlass, you can build the second Gunblade if you want but the 2 actives that both slow are pretty helpful. Once you have these 4 items, buy the Red and Green Elixir and you should be able to solo baron at 20 minutes. The last 2 items you can ether get more survivability items. I highly recommend a Frozen Mallet, it gives you more AD, and more HP. The last item you can should get would depend on the other team, if casters are messing you up you might want to get a FoN next, or if DPS/AD are hurting you, you could get a thornmail.