Not Updated For Current Season
Start off buying Doran's Shield and a health potion, you certainly can get a kill early on but it is rather hard before you have your debilitating poison and a few ranks in Expunge.
As soon as you can build berserker's greaves, do it, the 25% increase in attack speed is very nice as it lets you get more stacks of poisons on the enemies making your expunge hit much harder. By this point you should be able to get kills on non tank champions rather easily.
Picking up the recurve bow gives you a massive increase to haste again, allowing you to get your full set of stacks on enemies very quickly. When combined with spray and pray followed by expunge, it makes you capable of holding off several enemies at once. By this point you should focus on ganking. Most enemies when they see how fast they're being attacked will run away, letting you slow them and continue to wear them down.
Finish making the madred's bloodrazor (which the recurve bow is a component of). Once you're to this point, tanks are added to your list of potential targets. This is where you start to really shine. Keep the red buff, the slow on it is very nice, as is the DoT.
Next, create your Infinity edge, this gives you more burst damage keeping your damage high enough to scare most opponents into running.
If the game is still going on, you can work on building two phantom dancers. With these you will have more than enough attack speed (really, you are well over cap, you're getting them mostly for the crit), and 100% crit.
Next you can build a black cleaver, to make your damage increase with each attack, but really you should never get that far.
At least in pubs, by the time you get your boots, bloodrazor, infinity edge, and are working on your phantom dancer, most teams will surrender. I have had 3-4 enemies try to take down a turret, only to be completely shut down and killed by a lone little rat.
This is my first guide, so be gentle. It's basically just a guide of what I found really works for me, both through experimentation, and reading guides on this site and others. Shout out goes out to Demok Omnibane for showing me the potential of Twitch.