Like the rest of the London crowd, Henry Lennox shows up only at the beginning and end of this book. He is an ambitious young lawyer who looks forward to climbing the ranks of London society. We can tell right away that we need to keep an eye on this guy. As the book says, "Mr. Henry Lennox stood leaning against the chimney-piece, amused with the family scene. He was close by his handsome brother; he was the plain one in a singularly good-looking family; but his face was intelligent, keen, and mobile" (1.1.56). Henry has always found a way to compensate for the fact that he's not as handsome as his brother. Instead of being pretty, he's smart and ambitious.
For all his great qualities, the most notable thing about Henry Lennox in this book is that he's totally in love with Margaret Hale. He even proposes to her in the early chapters of the book. When Margaret turns him down, he asks, "'Margaret,' said he, looking into her eyes, which met his with their open, straight look, expressive of the utmost faith and reluctance to give pain, 'Do you'—he was going to say—'love anyone else?' But it seemed as if this question would be an insult to the pure serenity of those eyes" (1.3.50). He's a sensitive dude, but also hurt by the idea that Margaret would love anyone else. Little does he know that at this point in the book, Margaret thinks she's too good for every man in the world. It's not him; it's her.
Even after Margaret has turned him down, Henry vows to keep loving her. He says, "Margaret, don't despise me; I have a heart, notwithstanding all this good-for-nothing way of talking. As a proof of it, I believe I love you more than ever—if I do not hate you—for the disdain with which you have listened to me during this last half-hour. Goodbye Margaret—Margaret!" (1.3.63). As you can tell from his language, he's a lot more poetic than a mill owner like Mr. Thornton. But at the end of the day, Henry just isn't the right guy for Margaret.