You seem to be talking about them more in terms of physical prowess. That concept might work with certain species, but with humans there's a bit more to it. A chimp might gain alpha male status by being bigger, stronger and more vicious than the other male chimps, but human males tend to do it by other means. Think about prominent men in our society. Is Tony Blair a hulking behemoth of a man, ready to rip his opponents into tiny pieces with his bare hands? No. Physically he's a puny little man. But brute strangth is not what he needs- instead it is his political and oratory skills have made him one of the most powerful people in the world.
If you define 'alpha- male' by the ability to attract a mate, again, it doesn't just come down to physical prowess or viciousness, although it can do. It can also come down to intelligence, creatvity, etc. There's an almost infinite number of things women are attracted to in men. The traditional one evolutionary psychologists point out is resources, so as to secure a safe environment for herself and her children. Is Wayne, who lives on a council estate, survives on the dole, drinks himself into oblivion every night and gets into fights and torches vehicles for laughs, the sort of man who is a safe bet, or has any resources? No. He's a witless, pathetic, repugnant sort of a man, with the IQ of a boiled potato. His staple diet of turkey twizzlers and 40 Lambert and Butlers a day make him fall somewhat short of a strong physical specimen too.
So to summarise, chavs are in no way alpha males. Taking society as a whole, they are the complete opposite, at the bottom of the heap. It is only within their own microcosms, the boundaries of which are the end of their street and the local pub, that they are anywhere approaching alpha male, and even then only over a certain few individuals, such as the unfortuante wife and children whom he beats into submission and are too scared to leave him, or the even more witless male morons who hang around with him in a pack as his subordinate.
P.S. In no way am I saying that all who live on a council estate live and act like this. I refer only to the tiny minority of pillocks who make others' lives a misery.