Well first off I would like to say I started of playing blitzcrank 3 months ago, it has been the champion I play like 95% of the time.
I started of by following this build:
Click.

This build was quite suitable for me, I noticed I liked the cooldowns (surprisingly I got rid of 'm in this build except for runes and the frozen heart in the end, but I never completed my build).

I totally believe the Archangel staff works great, it increases your mana, which helps on your shield and obviously the extra damage in combination with magic pen boots/runes is awesome.

So, in the introduction section I already explained why I use archangel staff and the Sorcerer's Boots.

So, why do I pick the other items.

I'll start off with the Rylai's Crystal scepter. This item grants you health, Ability power and has an awesome passive. Which really works pretty good if you ask me.
Whenever you grab, punch or ultimate(may it be passive or activated) you'll slow your enemy.
Since blitzcrank's main power is the awesome Crowd Control, the additional slow will make sure you enemies have hard times escaping you.

As for the last three items, those are totally situational based.
Personally I like warmogs cause of the awesome health and health regen. But if your opponents team has a heavily AD team, you might want to start of with thornmail.

Frozen hard seems quite obvious too. Extra mana, extra cooldown reduction. More mana = more survivability, less cooldown equals more crowdcontrol.

In this section I'll continue on items, except for this time I will explain why I pick the order. or even which order might suit you.

First off, the Sapphire crystal + the potions.
Sapphire crystal: Seems obvious, grants mana for more grabs/punches. And gives you the additional shield.
The potions: personally I prefer buying 2 health potions, this is since I noticed that I always have that one mana potion left @ 20 min. But this is something you've to discover for yourself.

Tear of goddess: great set up for the archangel staff which you'll be buying later. Whenever you are at base, make sure to use your spells whilst you're still regenerating mana. (the 4 sec delay seems bugged, so just spam 3 abilities grants you 12 extra mana). By the time I am ready to buy the Archangel Staff my Tear of Goddess grants me between 600-800 extra mana.

Boots of speed. Obvious starter, doesn't need an explanation.

Giants belt. This item takes a while to build, sometimes I buy my Sorcerer's boots before buying this one, but this depends on when I need to go back. This item is also a great set up for your Rylai's Scepter.

Sorcerer's Boots: This item ends up giving you great late game damage. If I end up solo lane I rather buy the Ionians boots for more cooldown.

After I got my Boots and Rylai's scepter, I finish building my Archangel staff, this will be my last damage item(not including Thronmail's damage)

Whenever you feel like you're short on money, or just not getting enough money to finish your build fast enough, try starting out with Faerie Charm or Regrowth pendant. And upgrade this to your Philosophers stone

Now I hear you wonder: "why the hell does he max out the Power fist if he doesnt have sheen/lich bane".
Well, the damage output of grab isnt something you should be relying on. The Speed boost from your overdrive spell is awesome, but you'll be able to handle being something slower early game.

The powerfist spell however has less cooldown every time you use it, this way you'll be able to have more crowd control.

Obviously you should upgrade your ultimate whenever you can.

Exhaust + flash.

Flash seems obvious. This will save your life(if you use it well.) and even will get you some kills. (flash in + ultimate, grab whenever they want back, power first, and well, you should have him by now, else it's not worth flashing for)

Exhaust, well I just love crowd control, use this near tower and your tower will end up with having another hit, which might get you the kill/assist you were waiting for.

I think I 've handled most of the points here(runes and masteries will come later I just have to find a way to make sense out of it xP)

Obviously, every opponent you'll engage knows you'll be hunting for that great grab.

BUT most blitzcrank players tend to stand still whenever they feel they are close to a grab. This way you'll be easy to counter by just going back behind a minion.

So whenever you play with or against a blitzcrank, make sure to watch your/his movement.

Furthermore, whenever you grab(whether you hit or not) make sure you walk back. This way you'll increase your grab range (early game this wont be as much, but late game with your overdrive spell your grab will pull them over quite a great distance.)

*note, Grabs like this(using flash while grabbing) wont work anymore.

I tend to play as supportive player. So I rarely get any minions. But even NOT getting minions requires a bit of skill/paying attention. after level 6 you'll be hitting minions thanks to your passive. Because of this you might want to stay away from minions.