Blitz is one of the most versatile champions in League of Legends, he can fill almost every role on a team.

AD/AP
This is where the magic happens!
When I play Blitzcrank I have a pretty tough time killing the minions so assists is where your going to make most of your money.
There are a couple of ways you can start with Blitz
1) My favorite starting items are Sapphire Crystal and two Health Potions this helps you stay in lane longer with more mana. It also helps with your passive. (taking 1 Mana Potion instead of one of the Health Potions is also a choice)
2) Another one I used to use was Regrowth Pendant and either a Health Potion or Mana Potion
3) I rarely use this one, but it works just fine. Take Boots 2 Health Potions and 1 Mana Potion

I like taking Mana Potions over Health Potions because I play more defensively in lane and try not to get harassed. One mana potion normally does the trick for blitz to stay in lane util the first trip back.

I almost always get the Tear of the Goddess on the 1st trip back so i can start stacking the passive, plus the mana regen lets blitz lane even longer.
If possible get Boots on the 1st trip back also, if not go ahead and wait for round two. On the second trip back i go for the philosopher's stone and boots. If you can't afford both just get the health regen and boots, third time back the Sheen is a must buy, becuase your damage is going to start falling off if you don't get a sheen. Finally finish off your boots to Ionian Boots of Lucidity.

Core Build = philosopher's stone

A reason that I love this build, because at this point in the build you have not yet decided whether or not your going AP or AD. This is the core build that i use, it gives you time to see how your team is doing and what the other team is getting for defensive items. After seeing that you can base your decision on what your going to rely on for your damage.

AD (Build 1)
So you have your core and decide to go for the common AD build, whats next? Stacks the mana faster than anything else, and the big chunk of AD it gives you helps that sheen proc. I play very defensive to make up for not building defensive and getting the

Blitz is a squishy killer. His job is to grab the carry's (ap or ad carry) and kill them early in the teamfights.
If you do this well and get the carrys to build a early defensive you've done your job. The defensive item that most carrys will get to counter blitz is a Banshee's Veil. If your AD. that gives no armor so no need for pen. If your AP, the banshee does a lot against you and you should probably just go for someone else. Also as AP you get magic pen from masteries and runes. If your tank you just CC the enemy team and let your team do the damage. I will also address this more in the items section.

Magic Pen Marks- this allows you to be able to play any type of blitz. (allowing to to wait and choose during the game.)
Armor per level Seals- I like the defensive runes on blitz because I don't get that many defensive items. I get the per level because it gives you enough the most armor. Mana regen is another choice.
Magic Resit per level Glyphs- same as armor. Cooldown Runes work too.
Quints- I prefer health quints to give an early game defense, because I go with scaling armor and MR. I've tried other quints such as Flat AD/Magic Pen/Flat AP/Armor Pen and i just like health the most. All of these quints work and are viable. I recommend trying more than one set out and see what you like the best.

The support tree has the most to offer Blitzcrank. The lack of CD items makes the support tree even more helpful. Make sure to grab Expanded Mind and Blink of an Eye. I get greed, but i also see mixing things up to get more into the XP mastery.

As my tank blitz i do switch to 9-21-0, because I grab more cd items and mana is less important as a tank. I do run 9-0-21 as a tank sometimes too, it just matters on what i feel like that game. Going 9-0-21 again gives me time in game to decide on what build my team needs.

1st I get grab that makes the most sense to help for 1st blood. Although the knockup wouldnt be terrible if you got it. The only thing that happens when you level up E is that it shortens the cooldown by one second. With early Ionian Boots of Lucidity I don't feel the need to level it up in the early game.

I max Grab 1st, because i feel as if its Blitzcranks main ability. It has a long cooldown so maxing it plus the CD from Ionian Boots of Lucidity makes it be usable to start the fight and once more in the middle of a fight then finally while chasing that final enemy.

Maxing the speed second allows me to get Ionian Boots of Lucidity over Boots of Swiftness. As AD blitz the attack speed from W and the early Trinity Force is enough to last the whole game.

After the point at level 2 I max E last for the super short cooldown. If you feel like it doesnt come off cooldown fast enough during team fights then you can always level a few points into before you finish W, but attack and movement speed helps out quite a bit more than a knock up a few seconds sooner.

[1 exception is that if your going AP and get fed for a super fast Lich Bane then start maxing E sooner so you can use the proc from Lich Bane more often.]

I use
Exhaust After you grab someone just exhaust to ensure the kill. Be careful not to waste it if your team is with you and can kill the person you grab without you exhausting them.
Flash I use flash on everyone. every game, every time. Its a good escape skill and in team fights to get to their squishes.
Other good spells
Clarity Blitz has a slight early game mana problem so this will solve that. Also his passive procs of current mana so it your low on mana and its about to proc go ahead and clarity to give you hopefully that edge.
Clairvoyance If your team lacks a support this is always a good spell. This also makes all those blind grabs that you try and make visible and easier.
Ignite Extra damage is always good but is it better than exhaust? Well that's up to you. Plus if no one else on your team takes it you might want to pick it up.
TeleportWithout Boots of Swiftness this adds to your mobility which is important to blitz.
Cleanse To help chase and run. Blitz is very fast and this gets rid of those pesky slows.
Fortify I don't like to use it but it's awfully nice to have someone on your team to have it and know how and when to use it.
Bad spells
Heal Its heal.... do i have to explain?
Ghost No need. Enemies wont be able to ghost away from you anyway.
Revive only used to troll
Rally see revive
Smite Blitz doesn't jungle

Blitz is one of the most versatile champions in League of Legends, he can fill almost every role on a team.

AD/AP
This is where the magic happens!
When I play Blitzcrank I have a pretty tough time killing the minions so assists is where your going to make most of your money.
There are a couple of ways you can start with Blitz
1) My favorite starting items are Sapphire Crystal and two Health Potions this helps you stay in lane longer with more mana. It also helps with your passive. (taking 1 Mana Potion instead of one of the Health Potions is also a choice)
2) Another one I used to use was Regrowth Pendant and either a Health Potion or Mana Potion
3) I rarely use this one, but it works just fine. Take Boots 2 Health Potions and 1 Mana Potion

I like taking Mana Potions over Health Potions because I play more defensively in lane and try not to get harassed. One mana potion normally does the trick for blitz to stay in lane util the first trip back.

I almost always get the Tear of the Goddess on the 1st trip back so i can start stacking the passive, plus the mana regen lets blitz lane even longer.
If possible get Boots on the 1st trip back also, if not go ahead and wait for round two. On the second trip back i go for the philosopher's stone and boots. If you can't afford both just get the health regen and boots, third time back the Sheen is a must buy, becuase your damage is going to start falling off if you don't get a sheen. Finally finish off your boots to Ionian Boots of Lucidity.

Core Build = philosopher's stone

A reason that I love this build, because at this point in the build you have not yet decided whether or not your going AP or AD. This is the core build that i use, it gives you time to see how your team is doing and what the other team is getting for defensive items. After seeing that you can base your decision on what your going to rely on for your damage.

AD (Build 1)
So you have your core and decide to go for the common AD build, whats next? Manamune Stacks the mana faster than anything else, and the big chunk of AD it gives you helps that sheen proc. I play very defensive to make up for not building defensive and getting the Trinity Force next. Then I get a Chain Vest and a Negatron Cloak to cover bot defense stats quickly. Then look at the other team and decide to finsh the Banshee's Veil or Sunfire Aegis next. And if the game is a million years long i like to finish with a Atma's Impaler.

AP (Build 2)
Bored and decide to carry your team as fun AP build? Then i like to get a Needlessly Large Rod for a boost of AP before getting the Lich Bane. Next I follow the normal AP and finish Rabadon's Deathcap into Void Staff. I finish Archangel's Staff next but i wouldnt blame you if you would rather go Zhonya's Hourglass because the active is extremely helpful.

Tank (Build 3)