Why did Sauron dare to publicly claim Middle-earth after the fall of Morgoth?

Basically, (1) he thought he could get away with it and (2) he probably saw himself as justified and maybe even a good guy, at least to start.

The Valar have been described by some critics as deus otiosus "idle gods" who create the world and then step back from it, leaving it to itself. (And this may show the Valars' good judgment, since their interventions in the world of Elves and Men don't generally go all that well...) Regardless, the Valar have a long record of having abandoned Elves and Men to their fate -- they let Morgoth rule Middle-earth for ages before capturing him at the arising of the Elves, and after he destroyed the Two Trees and then escaped back to Middle-earth, they let him take it over again and ignored him until, finally, Earendil was able to get through their barriers and beg their aid.

The Valar don't have a very strong track record at opposing evil in Middle-earth.

Sauron himself was a Maia whom Morgoth corrupted:

Of old there was Sauron the Maia, whom the Sindar in Beleriand named Gorthaur. In the beginning of Arda Melkor seduced him to his allegiance, and he became the greatest and most trusted of the servants of the Enemy, and the most perilous, for he could assume many forms, and for long if he willed he could still appear noble and beautiful, so as to deceive all but the most wary.

It seems possible that he was not wholly evil and, for a time, really wanted to "go straight" but, also, had too much pride to submit to the Valar.

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair hue again and did obeisance to Eönwe the herald of Manwe, and abjured all his evil deeds. And some hold that this was not at first falsely done, but that Sauron in truth repented, if only out of fear, being dismayed by the fall of Morgoth and the great wrath of the Lords of the West. But it was not within the power of Eönwe to pardon those of his own order, and he commanded Sauron to return to Aman and there receive the judgement of Manwe. Then Sauron was ashamed, and he was unwilling to return in humiliation and to receive from the Valar a sentence, it might be, of long servitude in proof of his good faith; for under Morgoth his power had been great. Therefore when Eönwe departed he hid himself in Middle-earth; and he fell back into evil, for the bonds that Morgoth bad laid upon him were very strong.

So Sauron may well have figured that the Valar, once again, had abandoned Middle-earth to its fate.

Seeing the desolation of the world, Sauron said in his heart that the Valar, having overthrown Morgoth, had again forgotten Middle-earth; and his pride grew apace.

I see hints that Sauron -- originally a Maia of Aule the Smith -- was in his heart a techie who kept looking at the world around him and thinking that it really could be run much more efficiently, and that he was just the Maia to do it. (Another sin of Pride, which seems to have been Sauron's downfall -- note that in LotR his pride more than anything blinded him to Gandalf's plan.) Note also that Sauron taught the Elves much about how to make rings of power.

But elsewhere the Elves received him gladly...for Sauron took to himself the name of Annatar, the Lord of Gifts, and they had at first much profit from his friendship. And he said to them "Alas, for the weakness of the great! For a mighty king is Gil-galad, and wise in all lore is Master Elrond, and yet they will not aid me in my labours. Can it be that they do not desire to see other lands become as blissful as their own? But wherefore should Middle-earth remain for ever desolate and dark, whereas the Elves could make it as fair as Eressea, nay even as Valinor? And since you have not returned thither, as you might, I perceive that you love this Middle-earth, as do I. Is it not then our task to labour together for its enrichment, and for the raising of all the Elven-kindreds that wander here untaught to the height of that power and knowledge which those have who are beyond the Sea?'

The Akallabeth comments:

In this Age, as is elsewhere told, Sauron arose again in Middle-earth, and grew, and turned back to the evil in which he was nurtured by Morgoth

So, whatever good intent Sauron thought he had after the second fall of Morgoth eroded with time and his pride increased and

Now Sauron's lust and pride increased, until he knew no bounds, and he determined to make himself master of all things in Middle-earth

So. We are now more than a thousand years after the second fall of Morgoth, and the Valar have remained in Aman, leaving Middle-earth to its devices and Sauron to his. Sauron looks around and says to himself, 'Well why not? They don't care about anything outside of Aman and won't intervene again unless I'm really foolish.' and launches his attempt to take over Middle-earth. And, at first, he may have told himself it was for Middle-earth's own good.