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Sauron was given the title "Sauron the Deceiver" because he used charm and deception to gain control over others. He taught ring-craft to the elves and gave Rings of Power to men and dwarves so that he could use the One Ring to bind others to his command.

Sauron was also called "The Deceiver" because he convinced Ar-Pharazôn, a Númenórean king of the Second Age to institute Melkor (Morgoth) worship. His deceptions led to the downfall of Númenór.

Here are a few quotes that describe his ability and willingness to deceive.

Sauron took to himself the name of Annatar, the Lord of Gifts, and they [the elves] had at first much profit from his friendship.

^{_{Of the Rings of Power and the Third Age}}

In Eregion Sauron posed as an emissary of the Valar, sent by them to Middle-earth ("thus anticipating the Istari") or ordered by them to remain there to give aid to the Elves. He perceived at once that Galadriel would be his chief adversary and obstacle, and he endeavoured therefore to placate her, bearing her scorn with outward patience and courtesy.

^{_{The Silmarillion V : Of the Rings of Power and the Third Age}}

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.

^{_{The Silmarillion V : Of the Rings of Power and the Third Age}}

Understanding he turned to subtlety in perverting to his own will all that he would use, until he became a liar without shame.

^{_{*The Silmarillion II : Valaquenta "Of the Enemies"}}

When Thangorodrim was broken and Morgoth overthrown, Sauron put on his fair hue again and did obeisance to Eönwë the herald of Manwë, 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.

^{_{The Silmarillion V : Of the Rings of Power and the Third Age}}

We know that Sauron wished to become a dark lord in his own right.

Though of immensely smaller native power than his Master, [Sauron] remained less corrupt, cooler and more capable of calculation. At least in the Elder Days, and before he was bereft of his lord and fell into the folly of imitating him, and endeavouring to become himself supreme Lord of Middle-earth. While Morgoth still stood, Sauron did not seek his own supremacy, but worked and schemed for another, desiring the triumph of Melkor, whom in the beginning he had adored.

^{History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Orcs}

The above quote says that Sauron both wanted to become supreme Lord of Middle-earth, yet also worked for the supremacy of Morgoth. Seems like his adoration of Morgoth could have been false just as friendship with the elves was false.

Given that he misled elves and men and dwarves, did he also mislead or betray his own master?

The only thing we know about Sauron during the First Age is that he captured Tol Sirion and held it until he was eventually defeated by Luthien and Huan. After that, he disappeared and kept in hiding. Possibly because he feared Morgoth's wrath, since Sauron's defeat had made it possible for Luthien to create the disguises used when she and Beren infiltrated Angband and stole a Silmaril. It's far-fetched to call this betrayal though - he was simply defeated.

So as far as we know, Sauron didn't betray Morgoth at any point, since he had neither a motive nor something to gain from doing that.