Sup guys, I'm WarZeeker and I've achieved Level 30 about 3 months ago, and since I scored 43/5/7 in Summoner's Rift (Classic) with Lee Sin on his free week in August 30 (was about lvl 28 at that time), I've decided to Unleash some of my builds on my favourites champions. Nevertheless, I thought my Lee Sin build is too good to release it as my first published build. So that's why Olaf shalt be the champion who I'll right about. I want to show 2 builds in Olaf, since every guide that has been written here just dosen't seem to me that they are quite good at all. There are some that are quite impresive, but I disagree with them. I can assure you, use this builds (map choseen has a lot of influence which we will discuss later) and you'll success with Olaf right away.
Ps: I know I will be talking alot about Atma's Impaler, but just keep on reading or get to the item discussion section, when I bring up B CLeaver.

Pros:

Lots of Damage output at all stages of the game
Fun to play with (watching enemies trying to escape thinking they ever had a chance to do so)
Unique abilities

Cons:

Is no good against assasins (LeBlanc Annie Lee Sin)
Needs pacience to know when to activate your ulti and ur vicious strikes.

Marks / Quintences: 9x Desolation

Armor penetration is what all of this, first build is actually about. Olaf's huge damage output comes from the fact that with his items, runes, abilities and masteries combination grant him. This is why: first of all, as u all know, armour reduces incoming physical damage by x%. If your armour penetration is bigger than the targets armour then you deal 100% of your attack damage in basic attacks, or 100% of a ability which deals physical damage. The point here is trying to allways be sure Olaf's dealing full attack damage. This is because, since you have 3 poitns at Havoc (1.5% more damage) and executioner (6% more damage when target is below 40% health), means that Olaf will actually will deal 7.5% more damage!!! WTF OP!!! Now that's the reason why armour penetration marks and quintences. Especially at the beginning of the game, it helps u farm and deal almost full physical damage to even some champions. It will help you get first blood and last minion hit better. Get them.

Seals: 9x Vitality
This means that now your harded to kill, and also it gives Olaf attack damage. This is because since your getting Atma's Impaler (+ 2% attack damage according to max health) and Vicious Strikes (1% of attack damage according to max health) plus the fact that u have 150 more health in masteries (at Level 18) means more attack damage. People also say, why not Armour? Physical carries have alot of damage output. But Olaf is great against people who dont have alot of burst damage. I picked per level, since I take Atma's late game, and vicious strikes too.

Glyphs: 9x Shielding
As I said before, Olaf Isn't good against assasins. This is because Olaf's greatness comes from his capability to deal tons of damage without dropping his health bar below 20%. This is because, if your patient enough, you will activate Vicious Strikes + Ragnarok + Youmuus combined with his passive, and no carry, fighter or whatever they are, will ever be able to get your health bar down. I'm not talking about killing you, I'm literally saying that they wont be able to drop that health bar down. Except for an assasin. This is because, since they have alot of burst damage, they can drop that (when u risk to be at 20% health to have alot of damage output with Olaf's passive) health bar to 0% in 1 second. This is why I pick magic resist per Level, to keep them from doing that less effectivly. Per Level since they are more effective end game, which is when u have everything which is neccesary to risk yourslef to be at about 20% health.

Offensive:
Summoner's Wrath: as I said before, Olaf is all about armor pen. Since Exhaust makes the target's armour get reduced by 10... do I really have to keep on writting about why to get it?
3 attack damage. A little more damage is always good.
6% attack speed. Nicee.
10% armour pen. For those rammus mother fu$/*% who keep stacking armour up to about 400 armour (na just kidding, I love rammus, his my main tank).
4% critical chance and 10% critical strike damage. Loving it.
1.5% more damage. Never been happier.
6 armour penetration. More armour pen fu*$ yeah.
6% more damage when targets are below 40% health. OMFG this is gona get nerfed.
Deffensive:
6 more magic resist. Nice against mage assasins.
2 more armour. Nothing impressive here.
108 Health at Level 18. More hp and more damage.
30 flat health. Nice early game, and again, more damage.

Passive - Berserker's Rage: For each 1% of health Olaf's missing, his attack speed increases by 1%. Massive attack speed boost when in low health.
Q - Undertow: Skill shot that range can be modified depending on how far the mouse is from the champion. Slows enemies. This is a great way to harras your enemies, mostly by the fact that if the axe is picked up, the cooldown is reduced by 6 seconds. Try to use it to hit multiple minions when they come all lined up.
W - Vicious Strikes: Gives flat attack damage, life steal and increases even more attack damage by 1% of Olaf's maximum health. You should take notice that warmog's farmed gives another 450 health, mining that he ends up with about 3900 hp, and more ad.
E - Reckless Swing: Deals tons of true damage at the cost of health. It is the nightmare of squishy targets and low level, low hp targets like TF MF Ezreal Cait etc.
R for Ragnarok: Olaf removes all disables and cant be disabled for 6 seconds. during that period incoming damage is reduced by xflat. Passive: xflat armour penetration. Summarized: You can go kill your enemies with no worries at all. It is very likely you score a pentakill. WARNING: might get a little bit borring killing the wholoe enemy team by yourself.

I take flash and exhaust, since Olaf has no blinks / teleports / whatever you call it, so that his chace isnt interrupted by a wall, or his escape (not likely to escape with Olaf since people fear you). Exhaust... I have already explained, more armour pen and gives a nice slow debuff to kill enemies easier, or even make Olaf and expert against other melee ad fighters (since they deal less damage).
Other Options:

Ghost: remove flash, yoummus ghost blade and then add a Black Cleaver in its place.
Ignite: Remove exhaust, since the damage out put will be recovered by ignite, nice true damage combo with Reckless Swing.
for this build, I recomend no other Summoner spells.

Doran's Shield: Nice item to open up with, gives health, armour and health regen, which is great for you to spam your Reckless Swing.
Berserker Graves: Since I do not pick any other items that give me a bonus on movement speed (without taking accout yoummus since it is an active), olaf needs boots. I do not recomend mercury treads, since the tenacity is wasted, and Olaf hasnt alot of attack speed at full health.
Youmuus Ghostblade: following the basics of this builds, I've pick Mostly items which give armour penetration. Therefore, I chosed yoummus also because his active combines with his damage output activation combo. If you do not like this item then do not get the boots either. just get 2 phantom dancers.
Starks Fervor / Blood Thirster: Life steal, attack speed, and armour reduction, which also helps other ad team mates. If you feel like you have alot of armour pen, get blood for more damage output and more surviability (more life steal).
Infinity Edge: Olaf needs some flat damage, and this is the item your searching for.
Atmas Impaler / Black Cleaver: Okey here's something new. I've never brought up the Cleaver up untill this point. This is because I recommend more atmas than B CLeaver, and because I had forgotten about this item. It is a quite nice Option. Now that I have thought about it, B Cleaver suits my build better, so do the following: get Cleaver if you still need more armour penetration, and get atmas if u need some armour and more ad.
Warmogs: This is a non negotiable item. Just get it. Tons of hp meaning more damage again.

In a few weeks I will be postin a video about how to play Olaf. Also I will explain why does the map picked to play matters a lot.
Hope you Liked the guide and try not making people cry about Olaf with this build being OP.