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.
First off, many people like giving Blitzcrank loads of mana to increase his survivability, but I have found more life/armor/resist makes Blitzcrank survive more in normal games.
This build focuses on survivability and ganking. Blitzcrank has Ghost and Flash which makes him good for chasing down other champions and running away from them. His items support his survivability by giving life/armor/resist, and his items also support his ganking by slowing his enemies down per hit. His runes boost his movement speed and attack speed, increasing his ability to gank, and also boost dodge and magic resist to increase Blitzcrank's chances of escaping danger.
For me, this skill sequence has been the best for getting early/mid game ganks. Rocket Grab + Power Fist = Early gank. Overdrive is nice early game, but Blitzcrank should be fast enough to chase down enemy Champions. The reason for maxing out Power Fist is simple; it decreases the Cool Down, allowing Blitzcrank to spam it and handicap another champion. After maxing Power Fist, max out Overdrive. Now you can practically outrun every champion, and your attack speed will allow you to kill Creeps faster too. Next, max out Rocket Grab to decrease the Cool Down, and although maxing Rocket Grab can help early, it costs too much mana to be beneficial. Finally, put points into Static Field any time you can, i.e. 6,11,16. The passive is great, and its activation does a fair amount of damage + silences enemy champions.
Greater Mark of Alacrity
This allows Blitzcrank to hit more often, and he will eventually slow his targets when he has Phage or Frozen Mallet.
Greater Seal of Evasion
This gives Blitzcrank higher dodge, which is for dodging damage and his mastery gives him a boost of speed when he dodges.
Greater Glyph of Shielding
This gives Blitzcrank Magic Resist to reduce spell damage, thus giving him a better chance at surviving magical attacks.
Greater Quintessence of Swiftness
A boost of speed is always nice for any champion. Blitzcrank benefits with this early game and gives him a little speed advantage over his opponents later in the game.
This is good for almost every game. However, if you're facing an AD heavy team, I would suggest Ninja Tabi to get the extra dodge chance.
This is not a "must", but it does help Blitzcrank gank/run from his enemies, as well as giving a nice health boost.
This is also not a "must", but in most games there are characters, like Ryze or Lux, who have the ability to freeze others in place. It is also good for absorbing unwanted damage. If you don't have to worry about burst AP damage, then I would recommend Force of Nature, which gives a big boost to Magic Resist and Health Regen.
20% Cooldown Reduction + 99 Armor is amazing. It adds protection against AD and gives Blitzcrank the ability to grab/smash champions faster.
Most say this is a must for Blitzcrank, and overall, it is a very nice item for Blitzcrank. However, if this is not cutting it for you, get a Phantom Dancer, which gives more attack speed and movement speed, increasing your chances of getting kills.
This is an overall good item for defense, giving several decent bonuses. It's passive is very good for trying to escape a gank, and its active is going for escaping or ganking. If you don't have a big problem with melee, or you have a big problem with casters, then grab Force of Nature instead.
Ghost and Flash gives Blitzcrank high survivability, along with ganking ability. Dash really helps early and late game. Flash allows you to "Tower Dive", as well as escaping a Gank Squad.
Most people go down Utility, but I've found Defense gives great survivability, and gives him tank-like qualities.
In my opinion, you can tank Blitzcrank out, which is very successful and what this build somewhat does. You can completely tank him out, but I find he can't do too much tanked out, other than annoy the other team, which can be pretty useful. That is why I made him tankish.
Blitzcrank is also good for Team Fights or Team Ganks. His ability to grab enemies and punch them in the air is perfect for targeting a single champion. And his ability to increase his attack speed and run speed makes it perfect to chase champions down and provide support to his allies. His ultimate is also very good for team fights, because it hits all targets doing moderate damage, as well as silencing them for a few seconds, just long enough to kill off a champion or two.