(N.B. In reviewing Hume's criticisms, it's interesting that they anticipate, in rudimentary form, contemporary worries for the argument (e.g., a posteriori necessities and the problems they generate for evaluating modal claims pertinent to the argument; and Peter van Inwagen's worries for the argument raised by the material composition debate).)
1st criticism: Any being’s non-existence is conceivable, including God’s. But conceivability is the only relevant evidence we have of possibility. So, any evidence we might have had for the conclusion of a necessary being is rebutted or, at the very least, undercut.
-Reply: God doesn’t wear his necessity on his sleeve, as it were. If we could just grasp his essence, we would see that he exists of necessity.
-Rejoinder: No. It’s clear that even if we could grasp his essence, we still would see that his existence is not necessary. For we can see that it’s true of every being, no matter what their nature might turn out to be like, that they could’ve failed to exist. For we can see that necessity is only a property of certain propositions (e.g., ‘all bachelors are unmarried’), and not of beings.
2nd criticism: But even if it were true that necessity can be a property of beings, and not just propositions, we have no reason to privilege God as the necessary being. For it could equally turn out that the universe is a necessary being.
-Anticipated reply: No. For we can conceive of the non-existence of the universe. And since conceivability is the only relevant source of evidence for possibility, we have good reason to think it’s possible for the universe to fail to exist. And if so, we have good reason to think the universe is not a necessary being.
-Rejoinder: Again, the same goes for God: we can conceive of each – God and the universe -- as failing to exist. So neither hypothesis about the necessary being has an epistemic advantage over the other. (But this point works to the advantage of the skeptic. For the proponent of the argument is offering it as an argument to take one from a state of disbelief or suspension of judgment about theism to a state of belief. So if neither hypothesis is more plausible than the other, then it fails in this task.)
3rd criticism: We can’t rule out – and the defender of the cosmological argument here allows – that the universe is eternal: it could be that there is an infinite, beginningless series of dependent beings, each one caused or explained in terms of the one that preceded it. But if so, then it makes no sense to say that such a series has a cause. For one thing can only be a cause of another if the former preceded the other in time. But there is nothing prior in time to a beginningless series of dependent beings.
4th criticism: If each dependent being in an infinite, beginningless series is caused or explained by the dependent being that preceded it, then nothing’s left to explain. To say that the series of dependent beings needs an explanation in addition to an explanation of each individual being in the series is absurd. For the series of dependent beings isn’t itself a being, any more than a collection of objects in a room is itself an object. One might call collections of objects like this, ‘objects’, but such “objects” go no deeper than linguistic or conceptual convention.