Starting items for jungle:
Those depend on your amplifiers, if you have/use Rapid Medusa Mask then go with Mask and Shield, if you don't have it then pick knife and shield, obviously take as many hp potions as possible and also consider adding 1/2 mana potions.

Stolen powers:
always take consume, it can be either AtGL's or nightmare batman's, that doesn't really matter.
For the second spell, my choose is psychic assault, it makes gank really easy. 

different choices could be:
Green Lantern's meteor drop -> basically you cast it and then grab your opponent to secure the dmg.
Joker's detonate -> personally i don't like detonate but i guess it could work
Doomsday's invulnerabilty -> this guide focuses on spell power AtGL so it's a bad choice imo.


AtGL is a melee AP bruiser, he shines durning mid game with strong teamfight presence due to his AOE spells. Unfortunatelly his skills have poor scalling so he falls off late game and by that time his major strenght is slowing enemies with passive.
Having in mind what i wrote above you should aim to finish game in mid/early late game.

Skill order:
Always start with W, even if you are battling early, AOE dmg is always useful and it adds nice stack of radiation, if you can proc passive on enemy its almost guaranteed kill.
Now for the second spell i pick grab, simply because i like to do lvl 2 gank with psychic assault, with a little help from your bottom lane it should be a kill.
About the passive, do not forget it have tremendous slow if you can proc it, being slowed by 50% is huge, use that to chase your enemies. Thats especially good if you are chasing and can cast ult on the enemy, as it is your only long ranged spell.
Use E durning teamfights to add one more RAD stacks to enemies and get a little more beefy.
Warning: If you are battling someone and he goes under turret, you can cast ult to finish him off but DO NOT FORGET, if he had few stacks of rad (for example 3), the ult will jump onto nearest unit (like minion) almost instantly.
Use your ult durning teamfights to deal huge aoe dmg and give them good ammount of RAD.

I think core items don't need explanation, they're simply to get you some HP and dmg. Athlantead Royal Seal is a must in every game, the slow is synergies ultra well with your skills.

Core items:
Most of them are very situational, like if you are ganking/pushing a lot then you can get logoz if you own Accelerated Mod. Some of them are to get you tanky, some of them to deal more dmg if you are getting fed early. 
Ra's Al Ghul's Robe is very good on AtGL, consider buying it as 5th item.

Auguments:
All of my auguments have CDR bonus, the second(or first) bonuses are either power pen or power damage. Always use atleast G/S auguments.

Jungle route:
Start at the bottom middle camp (the big one), call your teammates to aid you so you will lose less hp and gain few seconds, make sure they won't take your experience. Then march towards small camp, the most bottom one. After those 2 camps you should be lvl 2 and it's the perfect moment to gank bottom lane.
Come from behind and cast psychic assault, save grab for running enemy. Your major role in this gank is to hit 4 RADs to proc 50% slow.
After the first gank you should propably go back to base and buy belt, then continue jungling, when your consume is ready go for the top big camp.

If you didn't have opportunity to gank bottom, look for ganks elsewhere. If you have duo top then start with top jungle if you see higher chances for succesful gank.

If you ganked top/bot succesfully and you are at decent ammount of hp, do the EMP Robot with your allies, that may help in future pushes.

Epilogue: 
AtGL is a very fun to play character with interesting mechanics, but don't make mistake, he is not a blitzkrank nor Pudge, his grab range doesn't allow for big plays near turrets or engaging team fights.
 hope you enjoyed this guide and will enjoy playing Atomic Green Lantern just as I do. Thanks.