Q: insanely good. It can do a decent slow, and a pretty high damage output. Also, you can pick up your last thrown axe to get a faster cooldown for this, so it's good for chasing down someone (throw it right on top of them and when you go to them, you will get your axe again and have another slow+damage)
W: One of the reasons Olaf is feared. Gives insane amounts of lifesteal AND attack speed.
E: The finisher. Lightning strikes from above onto your target. This ability costs a little bit of health, but it is refunded if you kill the target. The most important part of this ability is the TRUE DAMAGE. This even lowers it's cooldown every time you auto attack! This means that you can do ~500 TRUE DAMAGE every 3 seconds!
R: Very useful. It basically gives a huge amount of AD and you become immune to ANY form of CC for a few seconds. If you're chasing down a low health morgana Morgana, and they root you, then pause to use tormented soil on you, Tormented Shadow you can just pop your ult, and the dreaded cc will be gone! This can also be used on defense if you are running away (idk why you would do that) and you get caught by some cc, you can jst ult and run away.

Early game for this champ is to focus on harassing and farming. You are one of the strongest early-mid game champions and should take advantage of this. Skill order is r->q->w->e. Harass with q, last shot with e, heal with w if you have lots of mana, and low health. If they get too close, you can fire off an e, and if they want to trade, use your w, as you will win. Be careful not to spam q too much because you can lose mana fast.

I think Olaf is one of the funnest champions to play, especially with the olaf 1v9 comic out. If played correctly, you can outdual basically anyone 1v1, straight up fight. However, if they are a good kiter, you might have some trouble,(someone like Vayne). If you frequently find yourself in a 1v5 situation with your team dead, do not fear! Olaf is here! The lower your health gets, the more effective your healing will be, and the rune, Last Stand will help you deal tons of damage. I've seen my friend play Olaf and with him being one shot away, he just activates his w, then hits a minion like 3 times and immediately goes back to 3/4 health.