You guys are just scratching the surface of Blitz. He has so much synergy with all his skills he can be anything but AP built. Any item that adds damage, attack speed, move speed and/or crit is welcome for this champion. He is one of the best initiators and chasers, but neither the best DPS nor tank.
You may not get all the kills on the team, but you will be a leading cause of winning team battles.
While solo queue'ed, I prefer damage items because it takes quite a bit of nuke to kill him so he usually doesn't get targeted first and I want the gold from the kills because I can't always trust my random teammates to carry the team. However, I usually play with a 5 man queue so I can rely on my teammates to play their parts as carry, tank, and AP damage so I can focus on setting up our team for ganks and causing havoc in group fights.
My Typical Item Build (works against most teams)
Boots of Speed (3)
Infinity Edge/Banshee's Veil
Infinity Edge/Banshee's Veil (which ever you did not get previously. Never get 2 Edge or 2 Veil)
Game Over...chose any item
Use the 9 to get nimbleness and the 21 to get mana, xp, ghost, flash, movespeed, cooldowns and summoner cooldowns.
Red = Critical strike (~8.5% base [I don't remember the exact # but it is about 8.5%])
Yellow = Dodge (6.75% base)
Blue = Cooldowns (8.1% at 18)
Quintessence = Dodge (1.5%), Dodge (1.5%), Crit (1.5%)
This totals ~10% Crit, ~10% dodge and 8.1% cooldowns
Blitzcrank can be the fastest hero in the game. I consistantly and constantly have 450 to 657 move speed at level 13. With Phantom, Ghost, Overdrive, Nimbleness, and Masteries he can move faster than any other champion. This is great for escaping or initiating. If an enemy uses flash to get away you can still get close enough to grab them within a few seconds. You could even stack move speed inside your rune page, but that would probably be a waste.
With the 30% or more dodge, tower diving/tanking, neutral creeping, and hero ganking/tanking is very easy.
Rocket grab is good for one thing, grabbing. It is not a significant source of damage due to it's high cooldown and cast animation so I only put 1 point into it. Attrition is the goal for Blitz when fighting enemy champions, not dealing damage. His fist keeps the enemy from escaping, attacking, and casting spells effectively. It's not very common for melee champions to try to 1v1 Blitz so when they get pulled by rocket grab they usually run. The slow from Phage is nice, but his fist slows and with his overdrive and rocket grab no one is getting away easily. Let your team get the items to do damage and get kills. Blitz is there to make it easy for them to get stacked with damage items.
The key to playing a good Blitz is to catch hero's off guard with rocket grab, defending your team's DPS heroes, and stopping the other team's spells. Carrying the team is NOT ideal for this champion. Blitz does not need items or neutral buffs to play his part. Let your team take the lizard and golem buffs whenever possible. Blitz does not need mana, slow, or cooldowns from neutrals with my build.
Extra HP is not important, but your team may need 2 damage sponges. If this is the case then replace Infinity's Edge with Phage or something else that adds HP and damage, or HP and magic resist.
Pulling a hero away from their team for an easy kill ruins the other team's chance of defending or pushing. Defending your DPS heroes is easy with fist, grab, and static field silence. You should not chase heroes down while the rest of your team is fighting, but chasing down that last hero with your move speed, rocket grab and power fist is easy (and extremely rewarding when you finish off the last champion for an ace). Getting a kill is not worth giving a kill to the other team. You want to push after every battle and to do that you need your team. Interrupting a Fiddle/Nunu like ultimate is essential for winning team battles. NEVER use grab to stop channeled spells right now as it will cause the spell to auto complete and deal 100% damage.
Use static field often. It has a short CD, good burst damage, and silence. The silence not only stops abilities but summoner spells as well so they cannot Flash, Cleanse, or Ghost. In team battles it is imperitive you silence their CC and high damage abilities before they are cast. As Blitzcrank, you should be the first hero on your team to use their ultimate. Even if you only hit 2 or 3 people with your silence it will wreck the other team's synergy, damage, and ability to take control of the fight. By the time silence wears off they fight will be mostly over for the other team and your team should be left standing, ready to push for the win.
I have developed special combo moves with Blitzcrank that I did not cover here, but I can assure you that when you learn of these things I was probably the first to experiment and fine tune them.
Thank you Riot for giving me such a blessed, OP champion.