Serious, Critical, Exploratory, Persuasive
Philosophy doesn't have to be stuffy, but morality and justice are pretty deep subjects and Hume isn't gonna treat them lightly. He'll let us in on his personal views sometimes, using language like "I believe" and "I say." Overall, though, the tone is what we'd expect from a philosophical enquiry.
Hume lays his cards on the table early on, declaring,
Men are now cured of their passion for hypotheses and systems in natural philosophy, and will hearken to no arguments but those which are derived from experience. It is full time they should attempt a like reformation in all moral disquisitions. (I.10)
See? This guy means business.
Hume sees other philosophers as having dropped the ball in their accounts of moral theory, arguing that
An opposition of the greatest consequence could prevail between one system and another, and even in the parts of almost each individual system; and yet nobody, till very lately, was ever sensible of it. (I.4)
It's one big mess and that's why Hume steps in. He's not in the mood for jokes—his goal to clear up the confusion. To quote the man himself:
What need we seek for abstruse and remote systems, when there occurs one so obvious and natural? (V.II.2)
Taking a critical angle, Hume points out that other folks' theories are far from perfect: modern philosophers, for instance, have harped on about virtue and vice, "yet have commonly endeavoured to account for these distinctions by metaphysical reasonings, and by deductions from the most abstract principles" (I.4). In short, these guys just aren't in touch with the real world.
Another thing Hume criticizes is the idea that sympathy and kindness are just fronts and that, really, we're all selfish. Hume's argument is that we have a natural sense of sympathy and that "everything, which contributes to the happiness of society, recommends itself directly to our approbation and good-will" (V.II.2). Unlike some, Hume's not a complete cynic.
Even though he's convinced of his theory, Hume knows that readers aren't gonna accept it just because he says so, and he recognizes "that nothing can be more unphilosophical than to be positive or dogmatical on any subject" (IX.I.13). Just because someone's sure of themselves or arrogant doesn't mean they're right. The word "enquiry" clues us in on the book's tone, and Hume goes on to explore how our sense of morality is formed, using examples to back up his claims.
For instance, Hume considers situations where justice is suspended and asks,
Is it any crime, after a shipwreck, to seize whatever means or instrument of safety one can lay hold of, without regard to former limitations of property? (III.I.8)
Similarly, after explaining the importance of sentiment in morality, Hume asks, "what else can have an influence of this nature?"
In case you haven't noticed, these are rhetorical questions: Hume's aim is to persuade us that the answers are obvious and for us to respond, "yep, this guy's right."