Sorry if this is nonsensical right now, its just a rough draft and I will organize my in depth information in time to come.

Blitzcrank can have many purposes in lane, ranging from going a tank, to a utility, to an outright troll AP support. However, the most effective and useful builds in my experience are those which prioritize peeling for the ADC, such as Lulu. One of the main reasons why Blitzcrank retains it's usefulness even through mid and especially late game is because of his incredible ability to peel for the carry (ADC). The best method of winning is to abuse Blitzcrank's early game strength by constantly landing pulls, getting kills, and getting your ADC fed, while at the same time warding so you do not die any useless deaths to the enemy jungler. Once you get your ADC fed, which would be at around 15 min, you should have your ruby sightstone at about that time. If the enemy laners you are against are more passive and not very strong such as janna and jinx, you can go roam mid, using sightstones to ward off the enemy junglier from ganking your lone ADC. However, if you are against an extremely aggressive comp such as nautilus/karma and draven, DO NOT LEAVE LANE. No matter how fed your ADC is, it is nearly impossible to 1v2 against a comp like that. Just keep this process going until you and your ADC have minimal deaths at mid game, and your ADC is fed. Once you reach mid game, it is the time when your ADC is able to start popping off and roam mid for some juicy double kills.

When is it the right time to roam?
It is suggested to roam once you get your tower or it has reached 20 min.

Once you enter mid game, your usefulness is limited to two criteria. How well you land pulls on carries, and how well you peel for the team. As the game progresses, however, you should start focusing less on landing outplay hooks, and rather focus on peeling for your ADC. There have been far too many games thrown by blitzcranks who have misspelled an alistar who just demolished the fight. It is preferable to not take the risk and stand on top of your ADC no matter how good your hooks are. Mind you, at this point you will probably have a "Face of the Mountain, Locket, and a redemption" which are all extremely strong at keeping your ADC alive. Also, by standing on top of your ADC, it gives an easy e for you in case a sin xiao or any other dash champions happen to rush your ADC.

You may be thinking, "what will a measly e do for my ADC?"
An e can transition into a point blank q. This combo can stun the enemy for an estimated amount of 2-3 seconds which is well enough time for your ADC to kill him. If not ,just ult exhaust, and pop some of your shielding items. You MUST keep your ADC alive which is fairly easy for a blitzcrank late game because of all his cc, but if you do manage to disgrace yourself and not save your ADC, either move onto the next carry, (mid/top lamer). However, if there is no other good damage dealer, gtfo of there cause you're useless. You would just die peeling for a full tank malphite.

What if my ADC is braindead?
The odds of you getting a braindead ADC in silver+ elo is very low and even unskilled ADC's can be carried by your awesome support play. As long as they can point and click, you can peel the enemy off as they do so. Trust me, I have seen braindead ADC's and they are extremely rare to the point where I would win 20 games in a row playing like this.

Mind you, Blitzcrank is not an easy champion. Your skill shots have to actually be on point and you need to be able to control the map. Many believe that the support have minimal participation in LoL, but you are the main player declaring objectives and what to do next because you have a lot of power in your team. Supports have to do one of the most thinking in a game unless you play braindead and just want to get carried. For example, if you predict a future team fight, ward up the location/objective such as baron so that your team has the upper hand. Also know when to ping the right objectives at the right time. Do not go for dragon when they are aced with an inhib tower up. Always get the inhib tower and inhib then rotate to dragon. Even if they have 2-3 still up, you are blitzcrank. You have a pull to easily pick them off. If the enemies really do not know how to play against a blitzcrank, they will keep walking into his hooks one by one so you can siege to their tower.