If you're not that serious about playing Twitch, or are just looking for a surface level guide to him, everything you need to know is in the Item Builds, Runes, Masteries, and Skills sections. This chapter is devoted to explaining the nuances of playing ratman, because I really believe he's different from any other ADC. I expect most of you reading this guide are already decent ADC players; you can last hit, position, orbwalk etc well, but if you spend all your time bot lane farming and you reach lategame 0/0/0 there's really no reason to pick Twitch over say, Vayne.
The advantage Twitch has over Draven (other than the fact that Draven falls off and completely depends on catching axes) is the fact that people underestimate him. I don't mean that they think he's straight up bad (only a small percentage of...persons of lesser intelligence think that), but they see him as a squishy late game carry who's sole job is to melt the enemy team with his ult (which is true), essentially they think of him as a less mobile Vayne. Another misconception is that he falls into the "stealth troll" category of champs like Evelynn or Shaco. Twitch is a champion that I play solely for his laning ability, and stealth is really not a key component of his kit at all, at least in lane. I really only use it for surprise trading and obviously the attack speed buff.
Mechanics on Twitch are equally if not more important than on other ADC's, but Twitch has probably the highest trading/kill potential of any ADC in lane. Like I said earlier, think of Twitch as a Draven that doesn't fall off. You would never pick Draven just to farm passively; you pick him because you want a kill lane. If you end lane with anything less than 5/0, in my opinion, you've wasted a Twitch pick.
What sets Twitch apart from other ADC's with high burst/trade like Draven, Corki, Sivir, or Ezreal is that A. he doesn't depend on landing skillshots, and B. you don't have to capitalize on the enemy ADC's mistakes to start/win trades. Trading with Draven entirely depends on waiting for the enemy ADC to walk too close for a CS, or his support to land CC. With Twitch, the rule is: if they're within range of cask, and you land cask, you can trade, because 99% of Corki's, Tristana's, and Ezreal's won't spend 1/4 of their mana to jump away just because you hit them with a cask. They'll keep farming, get hit by your support's CC (hopefully), maybe even try to fight back at you (again, most people see Twitch as a squishy rodent), at which point you disengage, hit E, and come out on top. My personal rule of thumb for trading is if you can land cask (2 stacks) and AA twice for another 2 stacks, you can press E. Don't get hung up trying to get 6 stacks, but don't immediately expunge right when you land cask either.
Another key thing to remember when playing Twitch is his lvl 2-3 kill potential. I know this sounds like a bullshit, low-elo thing to say, but I've probably gotten a lvl 2-3 first blood in 80% of the games I've played with Twitch. I'll try to post a video of what I mean, but essentially what I'm trying to say is best illustrated by a Taric/Twitch lane. Nobody would try to trade with a lvl 2-3 Corki/Taric lane, but for some reason Twitch/Taric is just a lot less intimidating at those levels. If Taric lands his first stun and you get the enemy ADC down to 60/70% health, the next successful stun is a kill. It's imperative that you go for this early kill, because a first blood, or an early double kill is GG I'm not even kidding Twitch snowballs that hard.
What will happen (and this will happen exactly as I tell it every single time) is you land cask, Taric follows up with stun, you flash into their face, Ignite and AA as many times as you can in 1.5 seconds. The enemy ADC will immediately flash away once the stun is over (more so because they're intimated by your flash than that they're actually scared of dying to you) and then you just press E to secure the kill. It sounds dumb, but lets say you've AAed 3 times in the 1.5 second stun (so 5 stacks of poison total). At lvl 2 you should have ~20 AD from masteries and runes, plus another 10 from Long Sword (which I start with pretty much every game), and another 5 from your Ignite mastery, meaning your lvl 1 E does 159 physical damage, combined with 60 true damage from 5 stacks of poison, and 56 more true damage from what remained of your Ignite (which I assume has only ticked once in the 1.5 second stun). In other words, you have a margin of error of 275 combined physical and true damage in finishing the enemy ADC off after they have flashed away (which is about half their health if you're generous). The reason you flash in immediately when the stun lands is so that when they flash out, you are still in range for expunge. Also, obviously this is a lot harder to pull off on champions like Ezreal (doesn't matter because Twitch naturally counters him anyways) or Sivir (most other ADC's escapes will be interrupted by stun), or if you get hit by the enemy support's hard CC (you have 1.5 seconds of time to use your flash, try to use it for the dual purpose of keeping yourself in range for expunge AND dodging enemy HARD CC, slows don't really make a difference). Lastly, this strategy is directly countered by Cleanse (obviously) and is somewhat harder to pull off against ADC's that brought Heal or Barrier.
I apologize for the gigantic wall of text, but, again, this section is specifically for those who want to know about the in-depth mechanics of laning with Twitch. If you don't really care and just want to play Twitch as a regular ADC, you don't have to read this chapter at all. Hopefully I can get a video of me laning with a Taric so I can show you what I mean, but unfortunately I have a Mac and I don't know how to record LoL with this thing.
TL;DR Twitch good at trading so do it. Also kill people at level 2.
HOW TO DO DOUBLE GOLEMS:
This is a pretty important topic that should be touched on. There's a lot of people, especially at my elo, that do double golems when they shouldn't, or don't do them when they should, or just do them plain wrong. Unless you have a Soraka, double golems are usually only worth doing if you can make a lvl 2 play in the time period that you are lvl 2 and your lane opponents are lvl 1, i.e. Twitch + Taric/Leona: do double golems; Draven + Taric: skip double golems and just go hard at lvl 1; Vayne + Lulu: skip doubles because wtf are you going to do at lvl 2.
Now, finally, how to do them. I just decided to write this section on a whim so I'll get a screenshot later, but you want to be standing to the immediate left of the smaller golem (which is on the left), with your support standing to the immediate right of the larger one. When the spawn, both of you should aggro your respective golems, and you bring the small golem towards your tier 2 tower (through the little gap in the trees) and kill it, while your support brings the larger golem towards your tier 1 tower (through the little bush) and lets it reset. Once you have killed the small golem, you and your support should bounce the large golem aggro between the two of you (just like in season 2), until it is dead. I usually take the small golem for myself and let my support take the big one, but you can do the opposite as well (just make sure both of you are getting one of the golem's xp). Using this method, you should take virtually no damage from the small golem, and a total of 50-100 damage between the two of you from the big one (depending on how good you guys switched aggros).