Olaf is a superb jungler, ganker and chaser which will be highlighted in this guide. Jungling with Olaf is different in the way that he excells at low health jungling, yes, I said low health jungling. Olaf's passive is what makes Olaf's jungling superior to let's say, Warwick, He may be a safer jungler, but Olaf is a lot faster.

I run 9 Armor Penetration Marks
9 Armor Seals
9 Magic Resist per Level Glyphs
and 3 Armor Penetration Quintessences

Running armor pen runes helps in jungling with Olaf be cause you will be doing true damage at some camps and will be getting close at others.
Armor seals are for that extra survivability in the jungle
Magic Resist per Level are for that extra bit of tankiness if you haven't picked up a FoN or a Banshee's Veil toward the end of the game.

The reason you go 1/8/21 is jungle safety and buff duration.
You take 1 point in offense for Improved Smite, 8 points in Defense, adding both MR and Armor, and finally, 21 points in Utility adding into Awareness for level speed.

First off: Your core items you get without fail are: Mercury Treads and Wriggles Lantern.
The reason for Wriggles: Easy jungling, Lifesteal, Damage, and, armor in a very cost effective package.
Mercury Treads because they shorten CC and give Magic Resist, Some may argue that Merc Treads are unnecessary because of Olaf's ult, but I disagree. The reason is simple, is your ult always up?
Next is definitely Frozen Mallet, It grants Health, A bit of damage and most importantly a SLOW!. This slow can be devastating to the other team because it stacks with Undertow crippling the enemy and even with Flash or Ghost it is nearly, if not impossible to escape.
The next part is debatable, You can either get Sunfire for AD heavy teams, Thornmail for High Damage AD/Crit strike teams, FoN for Caster heavy/Bursters.
Finally the last 2 items for synergy Atma's and Warmog's, the deadly duo.
Warmogs's for the huge health increase and Atma's for the damage scaling from the vast amount of health you have late game.

With Olaf you start with the basic Cloth Armor and 5 Health Potions.
Start at Blue, you'll need a leash. Attack Blue, once you take damage pop a health pot, once it gets to about 400 health remember to use Smite, don't worry about cutting it close, Olaf is a low health jungler. Also, remember to try to hit all the creeps with your axe to maximize your jungle speed and safety.
Next drop a point into Vicious Strikes activate it while attacking wolves.
Move on to Wraiths, activate Vicious Strikes, Throw your axe and try to hit all of them.
Next is debatable, you can go attempt the take Red or you can go to Double golems. Either way after you finish one you should go back, buy boots and grab 1 or 2 health pots, Take red if you don't have it, gank if you do. Rinse and repeat.

Basically you want to max your Q, and some where along in there get 1 point in W and them max your E and take your ult whenever possible.
Maxing your Q first helps you take the jungle, has a really nice slow, does multi-target damage and is VERY good for clearing creep waves.
The one point early game in W is for the extra damage and the little bit of lifesteal that it grants.
Next is your E for that devastating true damage, basically it can be used to finish off anything.
Dropping a point in your ult when available is common knowledge.
Ragnarok is arguably the best ult in the game, It gives you passive Armor Penetration, Damage Reduction and you become Un-CCable for 8 seconds at max level allowing you to do whatever you want during that time.

Smite and Ghost, a must have for Olaf.
Smite for jungling easier and Ghost for ganking, escaping (Ghost + Ult + and axe to your chasers' face = Impossible to be caught.) chasing and initiating. Ghost is an all around good spell and should always be used on Olaf.

Olaf's role in teamfights is to lower the opposing teams health and to chase any runners, since he is immune to CC while his ult is active you should focus anyone who has hard CC and that is threatening to your squishies. Also, spam your axe when it's up and keep hitting the lowest health players with your E.

Special thanks to TheOddOne and Dyrus, this guide is based on TheOddOne's SoloMid guide and just watching Dyrus on streams I compiled this into a format that is comprehensive, yet simple I aim to make more guides in the future if you guys like this one.
PS: My first guide.

Update 1, July 14, Added: Teamfights section