Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
Okay so this is my first build, I have been playing Olaf for a while now and have found this build to work for me.
Now I start out with the 2x health pots and 1x manna pot for laneing longer. I then move in to the ninja tabi boots for they will stack with your masteries and rune book. After that I build up a frozen mallet not only for the health/damage from the item but the slow effect as well. (Olaf is really slow so you need that advantage) after frozen mallet, It will depend on what item comes next. If you are laneing against a Kenen or Ryze (that type of character) then it would be in your best interest to build the force of nature. Not only are you going get the ever so important magic resist from this item (that will stack with masteries) but you are going to get a little armor from it and health regen which is great. Okay now say you are not too worried about magic defense at this point. Fair enough stack 2x sunfire capes. the reason behind this thought process is that while you have a bunch of health yes, olaf has NO armor a jax or panth or hard hitting melee champ with armor pen is going to wreck you. Now stacking the 2x sunfire capes allows massive damage/survivability since you are going to be getting the health and armor from this item not to mention the bonus 40+ magic damage from it.
Alright now assuming the game isn't already over by the time you get your 3rd item we move to warmogs armor again this is for the mass health/healthregen on the item pretty strait forward. For the final item I would go with a ghost blade. The reason for this is that not only is Olaf slow at running but he is extremely slow at auto attack as well, so he relies on his ability very much. The ghost blade is going to stack with the cool down reduction from your runs and masteries and also giving you a little extra damage which will help with your axe throw damage as well. This item is debatable and can be thrown in at really any point you see fit I use it was my last.
For the summoner spells I have chosen I picked ghost (due to Olaf being extremely slow) this will allow you to catch an almost dead champ and end him. Or, if you are in trouble you can pop rangnurok and ghost and pretty much get out of anything/big trouble quickly.
As for ignite that's pretty self explanatory.
Now the build is a little slow going if you are not doing well on the lanes or have a competent partner. Now this is why at lvl 3 I get axe throw. Axe throw while it won't do a ton of damage it will slow them allowing you to catch them to wreckless swing them possibly ending them or allowing a partner to catch up. It is also a great harassment tool and can also get you a KB at low lvl. Also you can't always rely on someone to play a char that has a stun so relied on yourself. It is also great at running away from a champ slowing them while you continue to move on. That is pretty much it. GL and have fun out there!