Epistolary, Formal, Varied
Entirely comprised of letters, Les Liaisons Dangereuses has the styles you would expect from private correspondence about very personal matters. As the writers vary in education and sophistication, the quality of the writing differs. We probably see the writing style as stilted and formal at times, but that's how people wrote in those days—way more than 140 characters.
Cécile, for example, composes her letters in a literate but youthful style, chatting about her day to her good friend Sophie:
Fortunately, Mamma is better today and Madame de Merteuil is coming to see us with the Chevalier Danceny and another gentleman. But she always arrives late, and oh, it is very tiresome to be alone for so long. It is only eleven o'clock now. I have to practice my harp […] (1.14.2)
It's a practical, concrete style. Compare it with the showy prose of Marquise de Merteuil:
Well, I am willing to inform you of my plans, but swear first that, as my faithful cavalier, you will undertake no other enterprise until you have accomplished this one. It is worthy of a hero; you will serve Love and Revenge […] (1.2.1)
Her style is much more poetic and complex, like Valmont's. Valmont, especially, has a tendency to be over-the-top:
Then, if she wishes, let her say: "I adore you;" she alone of all women, will be worthy to utter these words. And I shall indeed be the god of her choice. (1.6.3)
Each of the characters has a distinctive style. As you proceed through the book, you'll sometimes be able to identify the writer by the word choice and sentence structure.