It's not a word I use myself, precisely because so many people consider it a synonym for 'poor' - which you're doing, right now. And as I said, I knew people growing up who were certainly not poor, but who would fit the description of 'chav' (even if we didn't use that word) on account of their appearance, tastes, attitude and behaviour.
Not sure what the Bullingdon Club has to do with all this. Sure, there's always been a tendency for toffs to behave like antisocial pricks. There's a great many people who aren't toffs but aren't on the breadline either - the great majority of people.
In fact the hatred of chavs which I can most understand is from other people who live in the same community.
Now you're onto something. The flipside of not all 'chavs' being poor is that most poor and/or working class people are not 'chavs'. Most of the kids I grew up with who had long hair, owned a skateboard and listened to punk rock - the very people the townies loved to yell homophobic insults at, chase down the high street and occasionally beat up - were from pretty humble backgrounds themselves. Wealth and income levels, in absolute terms, have less to do with it than you're making out, I think. The idea that it begins and ends with middle-class people disdaining working-class people is grossly inaccurate, anyway.