Firstly, let me say that this is my first guide and that I am not a conventional League player. For example I will, more often than not, neglect to buy boots and/or potions of any kind. Yes, they help a lot, but personally, I have no need of them early game, especially as Blitzcrank.

Blitzcrank can be built so many different ways, and to be honest, I haven't been able to have a full build before the opponents lose or forfeit. So I'm only going to show you the core items and depending on how the match goes, or how you play, you can build him how you need him. He's versatile like that!

I play Blitzcrank as an AP/support. Thus, I typically get the beloved flash with an exhaust, saved for after my victims finally break free of my pulls, knock ups, slows, and silences. Of course this is preference, and an ignite can guarantee the kills whereas a clarity will keep your mana filled.

If you're playing on Twisted Treeline, you have enough money at the start to buy a Tear of the Goddess. Doing so will give you the mana regen that you will desperately need to cast your Q more than 5 times before going back home. Not only that, but the passive ability of the Tear of the Goddess carries over when you finally buy your Manamune, so it's always a good idea to get it asap.

Boots! As I previously mentioned, I don't use boots with Blitzcrank, as his W can get me where I need to go. Boots are optional.

Potions! Those are optional too.

I would not recommend buying the Manamune first. The passive takes a while to build up, so you have time. Instead I would start buying the Iceborn Gauntlet. They are the key to Blitzcrank's potential. They give him everything he could possibly need: CDR, AP, armor, and mana as well as an on-hit AOE slow (Spellblade is the name) and some extra AP damage. Once you buy those gauntlets, Blitzcrank becomes an early game god.

After the gauntlets, the Manamune should be your second item regardless of the progress of the Tear of the Goddess's passive. The Muramana is just a fantastic way for your passive to really shine. I've killed many a champ who dove for my last bit of health only to be tricked by a mana shield of over 1k.
Your abilities will consume a few hundred mana each time, but with the Muramana, it hardly makes a dent in your mana bar, so go ahead and use that Muramana's toggle ability for some extra punch if your gauntlets aren't enough!

There you have it. Now your build is up to you! You have the core items, so now you need to build around the game. I will leave my suggestions for builds down below.

This is for after you have your core items and you're looking for what to build next.

Tank Build:

If you are just being focused, and you would really like to be a little more tanky, then this is for you! Your Iceborn Gauntlets and Muramana already help with your tanky build, so lets get some armor or magic resist. Lets see...:

A Ninja Tabi for boots
A Randuin's Omen for those pesky AS champs
A Frozen Heart to keep the distance when you need it, or to keep the distance closed in chase
A Runic Bulwark for the general tank things (armor/MR/HP) or Overlord's Bloodmail.

You know... typical tanky stuff. Additionally, you could do a Thornmail or Warmog's Armor if you're on the right maps.

Support Build:

How nice of you to feed your allies! What all does Blitzcrank need for this job?...:

Ionian Boots of Lucidity pull more often!
A Runic Bulwark to help protect your ally. I mean, you are pulling a champ right at them...
A Locket of the Iron Solari
A Frozen Heart (some of these things are repeating aren't they? oh well)
A Shurelya's Battlesong or Zeke's Herald or Will of the Ancients any would do.

The play styles of supports vary. From Blitzcrank, CDR would probably be the best support you can offer, but that may not always be true. I've left off the obvious sight wards and things, because this is (more or less) a guide for twisted treeline.

AP Build:

This is a build I rarely go into, but it is a viable option as both your Q and R have a 1:1 ratio. Your passive R has a 25% ratio, but it hits quick enough to do some damage!
We need a Wooglet's Witchcap
A Deathfire Grasp
Sorcerer's Shoes
and a Liandry's Anguish

You should always start out with your pull (Q) at level one. It allows for a lot of burst damage, and if you're laning with someone your grab will almost ensure a kill before the minions arrive.

Your second ability at level two, should be your knock up (E). There should be a wave or two of minions around by now, and if you can get the champion you pulled to stay put a little while longer, they will regret ever having to lane against you.

(Always remember: Q then E!)

The third ability at level three should be your W. By this time, the enemy champs are either staying away from you, or constantly ganking you, so you need a quick way to get away. Of course you can always use it to hit your poor opponent faster when you pull them close!

After that, I can recommend to you two choices: Q or W, as leveling your E gives you nothing other than a marginally smaller cool down.

If you're more of an AP builder, I suggest you max your Q first for that burst damage. However if you're playing a support, I would go ahead and max your W.

Of course your R should be leveled up whenever possible, and the cool down on it isn't much, so use it frequently rather than saving it up for a sure kill! If if comes down to it, it's a very nice minion cleaner.

Pros:
Pull, knock up, silence. If only he had a slow to top it off... (Iceborn Gauntlets)
Passive Mana Barrier... what more do you need there?
Can be a Tank/AP/Support either separately, or all at once.
Pull over walls

Cons:
Focused a lot.
Abilities use a lot of mana early game.