Hi, I decided to make this guide after me and a friend were trying to teach another friend a good build for Olaf, and after discussing some we decided to do this guide, because we got some good feedback from other people.

Here, I'll try to explain how to play Olaf, while I won't go too deep and explain counters and synergies, I'll give some good advices.
Olaf is a top laner or jungler(we'll focus on the top lane on this guide), with great survivability, chase potential and damage, what makes him and extremely good champion that will just ignore who else focus him at the teamfight with his Ragnarok and straight up kill or shutdown the enemy carry, and this build focus on helping you do just that.

Pretty simple, R > E > Q > W

I max E first for the increased true damage and good harass spell, because it has a decently high damage and relatively low cooldown. Then max Q second because it will be your way of pushing the lane after lv9 so you can help other lanes and/or destroy the top turret asap.

But, if going against a champion with REALLY high sustain, try maxing W first, because it will give the needed sustain while active for you to just stay in lane and don't fall behind, with the Philo Stone it will make it very hard to harass you out of the lane.
And if going against someone with great escape/chase potential like Rumble and Riven, max Q first because of the increased slow and decreased cooldown, so it will help you chase these annoying champions that hit and run much more effectively.

The standard build is that one on the item section, but you can try switching the order of Randuin's and Maw Of Malmortius, go MoM if going against AP, and Randuin's vs AD, also, if going against a heavy AD champion, like Xin Zhao, Riven or Gangplank, you can go Ninja Tabi and sell it later for your Mercury's Treads, because you will really need that Tenacity lategaem so you can do your job in the teamfights.

Pretty basic, go ghost ignite for that increased chase/killing potential, get flash if you really think it will be needed, but I think ghost is an overall better spell for Olaf than flash, because it synergises well with your ult and your Q combos when chasing.

Teleport is also a good choice it you want to split push or have that extra presence in lane that can sometimes get you ahead of the enemy top laner.
Exhaust is good too against high physical damage dealers.

Trading is a very important part of top lane play, dont let the enemy get damage on your for free, always try to counter it, or maybe let the minions aggro him for a while, it will push the lane on your favor and make them have a hard time last hitting under the turret, but when it happens, remember to ward the river so you don't get free ganked.

Focusing on having good trades is the reason I chose to spend those 2 points in reduced minion damage, so you don't get aggroed too hard and win the trade if going in even.

Killing the other champion is always a good thing, specially on top, it is the easiest possible lane to snowball from a kill, since most champions there are melee and after you get the kill you can just zone him so hard he won't get cs.

TL;DR: Focus on last hitting, harrass and go for trades when you think you can win, when you dont think you can and he comes to attack you, just EQ and back away. ALWAYS try to have your lane warded since a lot of junglers like to camp top.

Go 9/21/0 for the good damage increased and armor penetration and 21 on the defense tree to be tanky enough to go into the team and kill the enemy carry, pretty simple.

Just last hit, when farming under the turret, since you're not going to go any damage items at the early game, getting one or two levels on your W will make last hitting the creeps under it very easily, with W active wait for the turret to shot the melee minions two times and the caster minions one time.
When alone in lane, you can kill the wave by throwing and picking your axe up several times, it's not really mana hungry and cleans the wave pretty quickly.

Olaf is a fun and strong champion to play, that makes you feel like a man when you play it.

And remember, in teamfights: go for the squishy ones, the AD Carry is the highest priority, Ap carry and support are the ones you'll go if the ADC isnt reachable. And it's better to get yourself killed to let your carries alive than you staying alive and the rest of your team dying, unless you ace them and you're the only one alive, then it's an obvious positive trade for your team.