Learning the matchup against Blitzcrank is extremely important in my opinion. He is a very popular champion, and this is due to his incredibly brutal gameplay. If you do not know how to deal with Blitzcrank, you will get owned by him. He's not a champion you can just "do your best" against. Failing to properly account for his strengths will result in deaths, and this makes him a very popular choice. If you're struggling at all against Blitz, you owe it to yourself to keep reading.
You shouldn't use this as a guide to playing Blitzcrank, but if you're totally unaware of how he works, here's a general rundown of his threats.
First, Power Fist's cooldown is pretty short, and he can often land multiple Power Fists if needed to kill his target. Along with Overdrive, Blitzcrank is really hard to shake once you're in close range, and he can slam you up multiple times in order to confirm a kill.
Blitzcrank's ultimate is nasty, but the real problem is the silence. It is natural to Flash away from a bad situation like getting rocket grabbed, but Blitz can silence you to prevent any Flashing or using other escape skills. Combined with the right lane partner, Blitz is very hard to escape.
Blitzcrank fists hurt my face
Rocket Grab is Blitzcrank's most feared tool. You need to learn to counter it decisively if you want to stand a chance. Don't be the player who threw away 11 kills in lane from rocket grabs. I've been there. It makes you feel pretty worthless.
Rocket Grab has a very particular range and is blocked by enemy minions. Learn the range; you can watch the video above to get a good idea of how long it reaches in practical situations. Blitzcrank likes to hide in brush or fog of war to obscure his position when he grabs, so do yourself a favor and expect him to be in those spots. Don't assume he's in any one spot if you don't know; assume he's in every bush until you know for sure. Those who have played against Garen will be familiar with this mindset.
On Dominion, Blitz will often try to grab you from bushes in the jungle area and pull you into teammates. You can't ward here as you can on Summoner's Rift (wards help a lot in this matchup), so if Blitz is missing, be wary of anywhere he might be hiding. It takes one grab to kill someone and put you very far back.
In any situation where teams are nearby, a successful Blitz grab means you need to engage. It's not specifically that you need to save the person getting grabbed, but you need to punish Blitz's team members for it. If they're bunched up, it can lead to a very big win for your team. In one game, I got grabbed by Blitz only to see Shen dash in behind me and shut down the entire enemy team at once. Our team converged on the clustered-up opposing team, and I felt a little less bad about getting grabbed afterward.
There are some techniques for avoiding grabs, but they are hard to teach with words. There are a number of situations that make it difficult to dodge skillshots because you have nowhere to go, and you should avoid these whenever possible. A few are given in the champion spotlight video above, but it's really something you have to practice. Whenever possible, try to keep minions between you and Blitz even if this means you have to flee from an unorthodox direction.
I've been aware of what Blitzcrank does to inexperienced players for a long time because I've been there. When I was re-acclimatizing myself to Summoner's Rift a few months ago, I played a bunch of games with LilyPichu, and since I was bad at pretty much everything in SR, I ended up as support in a lot of the games (I also like support, OK?). In one game, Lily was playing Ashe, I was playing Sona, and we were fighting against Blitzcrank. Even though I knew about getting rocket grabbed, I pretty much fed the enemy team's carry and single-handedly lost the game for my team. Lily was really encouraging to me in voice chat, but I knew I was failing. I was apologizing for my fails and trying to figure out how I was constantly getting into those bad situations.
Let's fast forward a few months and a bunch of champion releases. I unlocked Graves recently and played with my duo queue partner in a Graves/Soraka lane. He's usually pretty bad at support, and my opponents ended up being Tristana/Blitzcrank. At first I thought it would be fairly easy, but this is because I was a total idiot. Tristana and Blitzcrank have amazing synergy, and these two actually knew what they were doing. Blitz would grab, and Trist would rocket jump at whomever Blitz grabbed, unloading her nuke combo and basically obliterating the enemy. This lane should have sucked. Tristana also did a good job of avoiding excessive push and keeping the lane fairly neutral, giving Blitz a wide area to grab people in.
However, Blitz was not as good a player as Tristana, so he was frequently baited. I have a pretty good head for how far Rocket Jump travels, and I would put myself in "bad situations" to last-hit while she was out of range of Blitz. He would grab me, and Trist would run to get to me, but Graves is pretty slippery and I'd be long gone. With her rocket jump and Blitz's CCs on cooldown, I would engage with my Quick Draw buff active and Soraka's armor buff in full swing. While it didn't get us any kills, we forced both Trist and Blitz out of lane several times.
They did catch me early, around level 5; Tristana dinged level 6, and they managed to combo me almost immediately afterward, giving a kill to Blitz, who had me Ignited, blocking my summoner heal as I fled with almost no hit points. However, I got low on health again near my turret, and a successful heal bait led to Blitz's dying to my turret, giving me a kill and putting me ahead.
By level 9, I was over 20 CS ahead of Tristana and actively able to bully both foes out of lane. I endeavored to cover myself from Rocket Grab as much as possible, and we managed to avoid any further deaths in lane. Healing and Infuse kept us going in lane while Blitz and Trist were frequently forced to retreat, as every over-extension by one of them meant pokes from Buckshot or someone's basic attack. The same harass from Tristana was easily handled with Soraka's healing and my lifesteal items. We eventually won the game by surrender; our mid was dominating, and the enemy jungler could not make anything happen (he died to several 4-man ganks trying to gank our mid).
The moral of this story is that solid defensive play is the best play against Blitzcrank. Obviously your strategy will vary depending on your support, but a "standard" support like Soraka, Janna, or Sona can win the lane for you as long as neither of you gives anything away. While I was lucky that our Swain had a dominating performance, that game could still have easily been lost if I gave kills to Tristana. Instead, I went on to ruin the enemy team in subsequent teamfights, feeding several kills to Swain and picking up a small handful in the space before our opponents surrendered.
The most important thing to do in Summoner's Rift when you're fighting Blitzcrank is to deny his lane kills. Other supports support, protecting the carry and playing a defensive role, but Blitz must get his carry kills or his lane will quickly get behind. Even Alistar, an aggressive support, can provide minimal healing to his ally and disrupt the enemies to keep them from last-hitting. Blitz is all-or-nothing. He's played even in top-level play because his utilities are amazing the mid and late game, but it's easy to take that midgame away if your carry is extremely far ahead on minion kills.
Don't get initimidated by Blitz, and don't get grabbed!
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