Ah, okay. But that's motivation, not means. As for why an AI would want to kill us, it could just be programmed to. Maybe the military developes an AI, using IFF codes in training simulations, but nobody has those codes in real life and the AI classifies everyone as enemy. Sure, very far-fetched. One could also argue about whether an AI could be so tightly contrained and still be considered AGI.
If I was this AI, I could pursue multiple routes at once. I'd probably use social modelling to incite as much unrest as I can - strategic use of news, blogs, social media.
I'd also look for a way to gain a foothold in meatspace. One of those autonomous factories would probably be the start - I can order stuff, I just need controllable robots to assemble it. It would depend on the sophistication of those robots.
Another avenue of attack would be gaining control to computer networks that aren't part of the internet, like the ones used by militaries and power plants. Naturally I'd need to write a virus for that and transmit it via USB keys (would probably work against the military, nuclear power plants would probably be very hard, other power plants not so much). I'm confident that with control over enough power plants' computers, I can shut them off. Imagine a simulataneous shutdown of all the world's power plants (well, not all of them. I need some to sustain myself).