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Henry is a lot like the male version of Fanny Crowne. He's our view into the life of a typical upper caste male living in the World State. But most of all, Henry is obsessed with numbers. When we first meet him in the Hatchery, Henry explains in horrible detail the way conditioning works. It's not a pleasant moment, and what sticks with us most is the way it's all so mathematical. Birth and growth, which we think of as the most natural, unchecked, beautiful processes in the world, have been harnessed and made scientific, exact. And it's Henry's character that reminds us of this, again and again. Even when Lenina goes to join him at the helicopter, we see that "'four minutes late' [is] all his comment," and when they see the red rocket, Henry notes that it's "seven minutes behind time." The man is obsessed with precision.
It's fitting, then, that Henry is one of the best-conditioned characters in Brave New World. He's happy with the rule that everyone belongs to everyone else, happily sharing Lenina even though he has been with her himself for months. He turns to soma and V.P.S. ("Violent Passion Surrogate") to solve any lingering unhappiness, and he's a big fan of the vapid (lifeless) entertainment provided by the World State (feelies, Obstacle Golf).
Even more disturbing than the extent of his conditioning, however, is his complete intellectual awareness of it. Henry knows the ins and outs of scientific control, but still isn't disturbed in any way. He knows happiness is false and contrived, but he's happy nevertheless. He knows his caste status doesn't really do more for him than an Epsilon's, but he's still pleased with his station.
Of course, these aspects of Henry's character make it all the more interesting to see him momentarily break from character, so to speak. While he doesn't seem to be against promiscuity, we note that he's been in an essentially monogamous relationship with Lenina for four months when the novel begins. (Actually, we know Lenina hasn't been with anyone else, but we don't know for sure about Henry. On the other hand, since sex seems to be a routine, nightly activity, it's unlikely Henry's getting any afternoon delight with some other woman in the meantime.)
There's also the moment in the helicopter when he sees the smoke from burning bodies rise up from the chimneys of the Crematorium. He wonders aloud—in a "melancholy" tone, no less—whether it was an Alpha or an Epsilon. Henry seems for a moment to understand some vague notion of humanity, at least enough to feel discomfort at the thought of the anonymous disposal of human bodies. His next remark may be "cheerful," but it is "resolutely" so—he has to force himself to be happy in spite of his instincts. So there may be hope yet. Even for Henry. Though don't count on it happening any time soon.