Olaf is a very powerful jungler is season 3, this is because his clear is very fast in comparison to other junglers and his ganks have high damage output and a strong slow if you can land your axe. Very few champions can duel Olaf in his own jungle or even in their own jungle. For this reason he is very good for invades.

Pros:
- Very Strong Duelist (early and late game)
- Your 'q' allows a VERY fast clear due to its 'zero' cooldown
- Has one of the best skins in game
-Pretty damn broken
-Has True damage
-Snowballs hard
-Good for counter-jungling
-Immune to CC with ult
- Has a pretty much constant slow if you can keep picking up your axe

Although Olaf is a very strong jungler, he does have his limitations!

Cons:
- Doesn't have a gap closer
- Relies the rest of your team to know what they're doing in teamfights
- Cannot protect the ADC very well (unlike junglers such as Malphite, Amumu, Shen and Maokai)
-If a player knows Olaf well, they can dodge your axe, move in a way where you cant pick them up.
- Without Ult up, ghost or Shureya's its easy to get kited

A lot of junglers in season 3 struggle with the early damage output from golems, but not Olaf. Simply start at wolves (with a leash preferably), progress to blue (again with a leash if possible), moving onto wraiths, followed by red, then wraiths again. You may sometimes find that by the time you reach red your smite is still on cooldown, this is why it is sometimes best to smite blue first.
After your clear you should have full health (or near enough) to gank for first blood. This gank usually comes some 10 seconds before the enemy jungler (depending on who they are playing).
Alternatively, Olaf can be a very good level 2 ganker, this is because your jungle clear ability is also your initiate on ganks, if you feel you are going to be ganking after Wolves ---> Blue ---> gank, then you may feel it necessary to take ghost instead of flash, or even exhaust if you still plan on building Shurelya's.

Passive: Beserker Rage - For each 1% of health missing, Olaf's attack speed increases by 1%.

This ability is fairly easy to understand, you lose health, you gain attack speed. But because of this your health potion management will be a little different during your jungle clear. Most champions need to chug a potion after wolves and after blue. But as Olaf, because your 'w' gives you sustain, i often find my first potion being drank at wraiths because i know my passive will give me a faster clear. As well as being good for jungle clears, this passive is particularly effective against turrets, if you know there are know enemy junglers around then ask your support not to heal you/don't drink health potions.
Q: Undertow - Olaf throws an axe to a target location, dealing (50/90/130/170/210) (+(0.5 per attack damage point)) physical damage to units it passes through and slowing them by (24/28/32/36/40)% for 2.5 seconds this slow then fades over time (fade due to a recent nerf!).

If Olaf picks up the axe, the ability's cooldown is reduced by 6 seconds.

Cooldown: 10 seconds
Range: 1000
Cost: 55 / 60 / 65 / 70 / 75 mana

This is my favorite skill on Olaf, having the cooldown reduced on picking up the axe is the main reason why your clear is so fast, and your dueling is so scary. Throw the axe the shortest range possible to pick it up and throw it again. After obtaining blue buff or some cooldown reduction items, its cooldown will be basically 0 if you pick the axe up straight away!
When ganking always throw the axe in the enemy's escape path, although it may not hit, the enemy will be forced to juke towards you allowing the lane to do damage and you to land reckless swing (see below).
W: Vicious Strikes - For 6 seconds, Olaf's attack damage is increased by (7/14/21/28/35) (+(0.01 per health point) [1% of maximum Health]) and he gains (9/12/15/18/21)% lifesteal and spell vamp.

Cooldown: 16 seconds
Cost: 40 / 45 / 50 / 55 / 60 mana

This ability provides Olaf with a lot of sustain in the jungle, like a 6 second Atma's this is very powerful late game and is what makes Olaf one of the best tanky bruisers. Because of it's vamp it also gives you the advantage dueling, as your health gets lower and lower, this ability combined with beserker rage (Olaf's passive) makes Olaf have huge dps.
E: Reckless Swing - Olaf deals (100/160/220/280/340) true damage to his target and inflicts (40/64/88/112/136) true damage to himself.

Cooldown: 8 / 7 / 6 / 5 / 4 seconds
Range: 325
Cost: 40 / 64 / 88 / 112 / 136 health

This ability is yet another reason why Olaf is very very strong when built with some cooldown reduction. This ability is very strong at any point during the game, especially because its true damage, in a long sustained team fight you can land this like 5 times which at level 5 is tonnes of damage, if you thought a Darius' ult was OP... this gets a 4 second cooldown even if you don't kill someone with it! "I'm OP? Your moms OP!" - Brolaf

R: Ragnarok - Active: For (5/6.5/8) seconds, Olaf is immune to disables and reduces incoming damage by (20/30/40).

Old Passive (now an active for 6 seconds while using ultimate): Olaf has (10/20/30) increased armor penetration.

Cooldown: 100 seconds
Cost: 100 / 75 / 50 mana

Because of this ult, Olaf is very powerful against teams that have a lot of CC specifically with the intent of keeping their ADC alive. When they initiate, just pop ult and assassinate the ADC. When i play Olaf, my main role in team fights is to zone high dps enemy players (usually the carry) so that my team can win the fight. As well as being great in team fights its also a really good escape mechanism (it is also why most run ghost on Olaf, i think both Ghost and Flash are viable, both have their strengths and weaknesses), just hit 'R' and you have an ultimate stronger than Master Yi's, hell throw a Kayle ult in there and you are unstoppable!

Sorry for making this chapter so long! there is just so much to say when it comes to Olaf's abilities!

Masteries:
I normally run 9/21/0 as i do for most bruiser junglers, but if you feel like your team needs a little extra burst, an assassin jungler build such as 21/9/0 is also a good option. Have a play around with the masteries i often find this is best for my clear and late game sustain.
Runes:
For most of my Junglers i run:
Flat armor Yellows
Magic resist per level blues
attack speed reds
and Movement speed quints

I do this as extra rune pages are expensive as are runes, in an ideal world i would run either armor Pen reds instead or attack speed due to Olaf's passive and reccomend doing so if you feel it necessary late game (even though cleaver and Ragnarok's passive is more than enough).
Like masteries i reccomend you play around with this to see what you feel is best (if you have the IP to afford all the runes).

My item choices may be considered a little strange to some, but trust me my choices don't come without reason. In my opinion every team needs a Shureya's (unless you feel you don't need it as you have a Malphite or something). By building it on Olaf it helps combat a number of issues:
1. The support doesn't need to build it and can instead focus on other items
2. You can replaces any machete building items as you already do good dps
3. It solves a lot of mana issues that i tend to have as Olaf
4. Olaf needs CDR as i have stated a number of times!!

You should always build situationally but i would always recommend starting with these 4 items:

Boots of mobility (with homeguard) - Gets you back into jungle quickly, gives enemy less time to invade
Shurelya's Reverie - Movement speed, movement speed, movement speed (and lots of other juicy things) [Note: delay building this if you feel underfed, just hold onto philo stone and you will soon feel the gold racking up!]
Warmogs - health is really good with passive, and you need to be tanky if you are running through the enemy team to reach their ADC
Black cleaver - I still think its OP even post nerf, plus you will near enough to true damage basic attacks to all those squishies!

Situational Items:

When faced with an underfed ADC/very little physical damage it may not be worth building a randuins.
If the AP mid is quite fed (and especially if they have an AP top or jungle) i like to build one of the following items to try protect me from being burst down:
Maw of Malmortius:
If i want extra AD and i'm against a burster or someone with an execution ult like Karthus
Banshee's veil:
Its passive is very powerful against high damage ult (e.g. Evelynn's is something to be careful of while building a lot of health). As well as this it gives health and resolves any standing mana issues.
Guardian's Angel:
There is not much more scary than an Olaf with guardians! Even if they are stupid enough to focus someone so tanky, you will just revive with low health, and your passive with be very powerful, plus mixed resists make you stronger overall!
Another option is a Spirit Visage, this item is again very strong against bursters, and will work very well with champions like Sona with lots of heals! (also your 'w' will benefit from it).

Ooooo i'm looking like a fed Olaf!!
I have had a number of games as Olaf ranked, where you pick up first blood and maybe kill the enemy jungler a few times (especially Shaco). Olaf can snowball pretty hard, and his ganks become even more terrifying, if you are getting very fed, it is sometime worth building a cleaver first followed by a blood thirster as the enemy team wont yet be doing a lot of damage (later in the game begin building health and resists such as Warmogs, Frozen Mallet and Randuin's Omen).

A lot of people like to build Ravenous hydra on Olaf, i would agree that it is a good item, but there isn't really enough chance in a teamfight to pop it constantly, especially if you are picking up axes, using other actives and popping reckless swing every 2 seconds!! If you feel you have the APM and feed to build it, by all means do so :)

Patch 3.01 saw the following changes to Olaf:
-Undertow:
Slow now fades over the duration

-Vicious Strikes:
Cooldown increased to 16 seconds from 12

-Ragnarok:
No longer passively grants bonus armor penetration
Now only grants bonus armor penetration while active

This makes Olaf's chasing significantly weaker, but it is still incredibly strong, currently considering a switch to swiftness boots over mobility to see if it can counteract the chase issues!

Viscous strikes is only really an issue if you run into an opponent after your jungle clear, my theory is after first clear, try not to use it when reaching wraith camp for the second time as this will mean its off cooldown for first gank.

This biggest problem with the nerf is the ultimate's passive now being an active, this means that you do a lot less damage than previously, recently i had started opting out of getting black cleaver 3rd and going for sunfire, but due to these changes its definitely worth a buy as you don't do anywhere near as much damage as you used to!

Feedback is appreciated, let me know if there is anything else you need to know about Olaf or jungling in general as i am happy to help/ point you to someone who can help :)
Many thanks to my ranked 5s team.

And the people who got me into the game:
Cairah
BulletAlchemist
Aelthi
and Sycsta for upping my jungle game :)

Hopefully will be adding specific match-ups in future. As well as alternative builds such as indepth tankier, and more damage builds.