How can I win this game as Zerg? First zerg gets harassed by a bunch of hellions, so you must get a couple of zerglings and roaches. When you tech towards those units, you have invisible banshee killing your queens/drones. So you must quickly tech up to mutas/hydras, but when you have 2-3 mutas finished and you take out those banshees, you now have MM ball with tanks in front of your base, because that terran was all the time massing these units. And because you were forced to tech in this early stage of game, you don't have much units. And GG is inevitable.
First off, if that's your replay, well done. I saw excellent micro in a number of places. For those of you who didn't watch the replay, he goes early gas into speedling, delayed expand. Gets bogged down by Hellion/Banshee harasses and eventually rolled by BioMech (I think I counted 4 Barracks).
Looking over the replay I saw two major problems:
I'll try to help address these both.
Early on you scout Rax straight into Factory, as a Zerg player this should be a tell tail sign for hellion harassment. Your natural response should be: 1) Spine Crawler at the Ramp, 2) Roaches. I did notice you eventually transitioned into Roaches but it was after the Hellion harassment started. Now its possibly you just underestimated your opponent, but I still recommend that Spine Crawler. The next thing to note is that 1:1:1 builds mean early Starport and that means Vikings. To protect your Overlords you're going to need Spore Crawlers, their range is just too long to protect with Queens alone. A couple Spore Crawlers will limit the range his vikings can attack from and save most of your Overlords. You do eventually get Spore Crawlers, but only after Banshee harassment has started. Even though you scouted his build I'm not seeing any adjustments prior to his units getting on the field.
Early harassment is very annoying to deal with and taxing on both your mental strength and your concentration; so its not extremely surprising that your macro started to fall by the wayside. You spent a lot of effort trying to shut down Hellions with Speedlings. While this is doable, its very taxing (especially if you fail). Its not that I think you can't shut down Hellion Harassment with Speedlings (because you certainly can), but you need to think of less costly ways to do it. You start to fall behind in the Drone count when his Banshees come out and you attempt to tech to tier 2. By the time he readies his final push you have a handful of Roaches and a dozen Zerglings against a strong BioMech push. Now to your credit the banshees do kill some of your drones, but largely you stop producing them. I'd say this is mostly the result of the extra pressure of the situation you're in. By the time he pushes he's out macro'd you so badly there really is nothing you can do.
So I've come up with a quick list of things for you to work on:
That last one is a little Zen.
And no matter what anyone tells you, if your opponent has Banshees and you're still in Tier 1, its OK. Try to calmly respond with a few Spore Crawlers and get back to your plan.
You need to scout more so you can detect what he does.
If he got banshees and you have no anti air it means you were already too far behind to win anyways.
Actually watched the replay, so I'll give some specific comments.
I see a lot of pros go 2 queens per hatch if they suspect possible air attacks. They would also help against the hellions. In this replay they didn't try to go up your ramp to your main's drones, but if they did, I've seen Idra wall off the ramp with 2 queens to deny the hellions. Another benefit of 2 queens is you have energy to spread out creep more which gives you both vision and movement speed. Also 2 queens can stay alive longer with transfusion.
Another thing I'd do is tech to lair faster. If I fast expand (I consider on or before 16 or so fast, not sure what other people consider fast) I'll go 2nd hatch before lair, but your 2nd hatch was around 20-something so I probably would've did lair first. It gives you the option to get an overseer when you saw the first banshee cause you know cloak is gonna be researching. Also roach speed is important.
I disagree on sending speedlings at the hellions because I saw about 5 of them. 1 or 2 hellions a surround would work, but you're just not gonna be able to get the surround of 5 fast enough.
That being said I agree with everything else @dbemerlin said.
I suggest instead of roaches use speedling to block ramp and make 3 spines and tech baneling/speed/mutas. Producing roaches when harassed that much may be too much in cost since the marine ball isn't massive at that point. If you get some extra lings check expansions doa bit of harass so the are forced to micro around more than purely harassing you. Plus against their push baneling kill off rines for your mutas to finish off mauraders/hellions/banshees. You may be able to scout it and do the one base muta tech build with spines/banes for the early harass that way your mutas are out when banshees are initially begin harass. Plus easy mutas may cause them to build up their marine ball heavier for banelings to wreck then you counter and hold them rines being produced with mutas.