Sayyid Qutb, an Egyptian, popularised a concept of Jahiliyyah (a simplistic translation of which is 'ignorance') in his book Milestones (which can also be translated as Signposts).
Qutb basically argued that most notional Muslims were no longer practicing Islam in the way that the first practitioners did, and that they had been corrupted (in large part by western culture). I don't think Milestones outright called for persons in a state of Jahiliyyah to be killed, but the idea that these persons were not real Muslims and hence not necessary subject to the same protections etc. paved the way for such action.
In terms of where Qutb and Milestones came from, it was in the context of the Egyptian Muslim Brotherhood and its frustrated attempts to acquire power essentially. Milestones was published in 1964, by which stage the Muslim Brotherhood had been sidelined by the secular Arab socialists (lead by Nasser). The government had driven the Muslim Brotherhood underground and the Islamists were essentially trying to figure out why they couldn't marshall enough popular support to tear down Nasser's state. So Milestones laid down an idea that essential the people weren't their allies, but the perhaps an elite group could lead them back to proper Islam. In a way it is a similar reaction a lot of political parties have when they fail to secure enough popular backing and/or are driven underground - it recasts the rest of society as something of an enemy or stupid (and by association, how smart/brave etc. are the followers of your idea) and that what is need is a small team of really committed people to lead them (making a virtue of their smaller base, and placing emphasis of taking action to resolve the 'crisis').
In Adam Curtis' documentary series The Power of Nightmares I think he claims that it was the later Islamic Jihad group that cemented the idea that persons living in a state of Jahiliyyah were fair game, and that idea terrorism might somehow shock them into true belief, but I don't know how accurate that is - I'm more familiar with the early Muslim Brotherhood.
Here is the documentary I mentioned. It doesn't deal just with Qutb, the Muslim Brotherhood etc. and Curtis does tend to over reach with his conclusions, but it is interesting never the less.