Whether this argument is valid really depends on what you mean by some of the key terms (this happens a lot in philosophy). But first: I don't think that a lot of philosophers promote skepticism. Most philosophers aren't skeptics, in the sense that they don't think that we have no knowledge about the world, or that we should doubt everything. But this gets us to the first term that needs to be clarified: what is skepticism? I gather from what you say later in your question that you take it to be something like the view that one should doubt everything. But then what does it mean to be a skeptic "about your own doubts?" One guess is that it means to doubt whether you should doubt, or to doubt that doubting is the right thing to do. If skeptics are people who doubt everything, that seems compatible with their also doubting whether they should doubt. There doesn't seem to be any contradiction there. We have doubts about what we think, and do, all the time. This would just be another instance of that: I doubt, and also I doubt whether I should doubt. This is especially clear if what we mean by "doubt" is just withholding belief (though I think you didn't mean just that by "doubt"). You can withhold a belief and also withhold the belief that you should withhold belief. This happens sometimes when you just have no idea what to think about some issue at all. So far, no contradiction. If a skeptic is one who doubts everything, then, it seems there is no problem here, with regard to having doubts even about one's doubts. Let's assume, then, that you meant to be talking about those skeptics who hold a theory, the theory that we should doubt everything. So if, instead, a skeptic is someone who believes that we should doubt everything, then that skeptic seems to be in an unstable situation: she believes something, and also believes that she should doubt that thing which she believes (since she believes that she should doubt everything). That seems problematic, as you point out. But you should also appreciate that most sorts of skepticism, at least as they are discussed today, are about a particular subject matter. Two prominent examples are skepticism about the external world and skepticism about morality. In both of those cases, the skeptic (the one who believes that we should doubt everything on such matters) is not in any trouble. She believes that she should have doubts about the external world and morality. But notice that this belief, that she should have doubts about the external world and morality, is not itself about the external world or morality (unless we unnaturally interpret "should" as a moral, rather than an epistemic "should" that concerns our intellectual, rather than our practical, conduct). So, her belief in skepticism is not something that her skepticism requires her to doubt instead.
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I've heard many philosophers promote skepticism. But it seems that skepticism is self-defeating, since the skeptics would have to be skeptic about their own doubts. Therefore, by virtue of that, they should not be skeptic. Is this argument valid?