Farm like a boss. It's so easy, but don't screw it up. His passive makes it easier. Level phosphorus bomb first for maximal farming and good harassing. A little later, a phosphorus bomb+missile will clear a whole back row. I love Manamune for superior farming like a boss. Starting Meki+2 HP pots. I don't start with it, but a somewhat early vamp scepter can be nice. Solo mid is your preference. If you have to side lane, go with someone who doesn't really need much farm. You may have a bad day against Heimer and AP champs, but you'll have a good day against pretty much every other range AD. Even if your first few levels go poorly, once you have missiles you'll have a consistent, very long range poke. Caitlyn's range may be imposing, but missiles immediately level that playing field. Don't stop moving. Never stand still for the time between shots. Unless you like taking skillshots to the face. I almost never have problems against any AD champ in the lvl6-11 range while laning is still going. Make sure you land Big Ones on champs. Don't feel the need to save one for a long time, but if it's up, hit the champ. Best done when they're farting around near a clump or minions and you can aim for those minions. Don't be too scared to Valkyrie your opponent if you have a decent health advantage and they have maybe 300-400HP. Punish them for sticking around in the danger zone. This opportunity doesn't happen all the time. Goes wrong sometimes on account of flash, etc. The idea is do it if you can kill them with a phosphorus bomb+auto-attack/missile. Otherwise, save Valkyrie for escape. Always keep enough MP for a Valkyrie.

If your opponent is lightly harassing you, make them regret it. Pop phosphorus bomb and an auto-attack when they approach to take a shot at you. Be aware when the aggression could be a sign of a gank or you're at low health or something like that.

Never stop farming.

Hit those missiles. Make them count. Don't suck at skillshots.

Yes, you have good armor shredding. No, you should not focus the tank.
On the subject of gatling gun, positioning can be rather difficult. I am no master at this. I treat it as kind of a nice damage bonus but don't get upset when I have to spin around, move, or generally can't keep it pointing at someone for the full duration. If the deal has been sealed on the teamfight, and the only thing in your way is that tank, it is very destructive. It's also nice to pop when clearing a huge minion ball.

My core in recent games has been Manamune, shoes (generally merc treads), trinity force. I like to start trinity with zeal. I like that % move speed. Sucks when you're getting outrun in your level 2 shoes. You'll probably want a BF item and defense item next in no particular order.

Coincidentally, I've had a few great 50-60 minute long Corki games in the last few days. All of them have been in the 250-300 CS area. End build generally looks like Manamune, Merc's (maybe berserker), Trinity, I.E., Bloodthirster, Defense item (Banshee's, Thornmail in one case). Really hardcore AD build since all the games were going quite well. Don't forget to buy elixirs when your build is done. Those damn Teemos I've been playing against are much less annoying when I have the expendable income to get oracles and clear shrooms as the game finishes up.

Are you still farming minions?

Previous general builds I've toyed around with involve getting the separate, stronger components of Trinity. Like I.E.+frozen mallet+phantom dancer+shoes+defense item. Dabbled with Stark's Fervor a bit as my AS item and went for a Bloodthirster for my big AD item. General AD builds work fine.