Jayce is one of very few champs who can fulfils any role from AD, MID to TOP , but i did discover his full potencial surprisingly in jungle. Not only he can clear camps pretty fast, but his epicness lies in his ganking power. With his range burst Acceleration Gate + Shock Blast following with the slowing-effect from To The Skies! + the overtime dmg Lightning Field and especially the knockback that can cancel Flash from Thundering Blow make him the most deadliest ganker of all champs.

Pros
+Fast clearing camps
+Gap-closer with slowing effect
+Best ganker with ability to interup Flash
+Can rape other junglers if they counter-jungle you
+Can make it to the late game
+Fun to play
Cons
+High skillcap champion
+Gap-closer can miss
+Teammate can be mad at you if you use wrong Thundering Blow
+Mana dependents

I take 9 Greater Mark of Attack Damage , 9 Greater Seal of Armor , 9 Greater Glyph of Armor and 3 Greater Quintessence of Attack Damage , because he get the benefit more from flat attack damage and flat armor and since i wont get lifesteal early on him so the extra armor from glyph make his jungling easier.

I chose the clasic 21/9/0.
In Offense i prefer to take Sorcery over Fury because you will need more the cooldown reduction than attack speed but you can skip Lethality to put it to Fury or maxing Sorcery if you want.
In Defense, take Summoner's Resolve , imo 10G everytime you cast Smite is huge. Bladed Armor will help you alot in early jungling and Veteran's Scars is a good improvement too.

Early game : Like other jungler, you begin with Hunter's Machete and 5 Health Potion ,some wards to counter counter-jungling and few hp and mana pot in case of need.
Mid game : Your core items are Madred's Razors, Boots Doran's Blade and Trinity Force(start building Phage first, Sheen second. As jungling Jayce , you will find that he's very mana hungry. If your mid are champs who dont need blue then you can grab blue everytime . If not , you can buy Tear of the Goddess to fix this problem.
Late Game :In Late game i find Jayce doesn't need Madred's Razors since you can clear camps very quick , so i sell this to make slot for better items. Rush Guardian Angel first, Last Whisper second, then you can decide between Bloodthirster or Black Cleaver depends on the situation. Last item to get is Muramana.

I start with Wolf(1 hit in Mercury Cannon, Shock Blast , change to Mercury Hammer , 1 hit the big wolf and To The Skies!) then atack the blue golem in the same order . take Thundering Blow in level 2 and start ganking (gank Top if you are in Blue team , gank Mid if you are in Red team). If you score the kill or assist , congratulation , you are now ahead of enemy's jungler and other lane. Continue with Wraith , now your Smite should be available again , so you do Red Lizzard and find possible lane to gank.
*note : Always gank in Mercury Cannon form to damage with the combo Acceleration Gate + Shock Blast and get speed boost to make more surprise.
Continue to farm if you canot find an easy lane to gank, also you can buy wards to counter jungling enemy because Jayce can practically own even Lee Sin if you can dodge his Q

To most people Jayce belongs to Top but imo he can only carries his team in late game in condition that he doesn't get shut down by ganks. But with Jayce in jungle , he can make surprise gank every time to help other lane to victory when you can get the same levels and items like you're sitting on Top to farm all game and hope you will not get ganked .
Replay
Replay 1 : http://www.leaguereplays.com/replays/match/2696735/
Replay 2 : http://www.leaguereplays.com/replays/match/2696746/
Replay 3 : http://www.leaguereplays.com/replays/match/2699866/

I hope this guide will help you find out the new way to own with Jayce, the defender of tomorrow.