Harry doesn’t actually appear all that much in the novel, but from what we’re able to gather, he’s the typical Victorian hero: young, attractive, active, devoted to his mother and lover, nice to children, good with horses, and blonde. (Yawn.)
He shows up for the first time just as Rose is over the worst of her illness, and we learn that he’s been in love with her for pretty much his entire adult life. Ignoring the ickiness of that (she’s only seventeen, and he’s twenty-five…was he in love with her seven years ago?), his devoted adoration of Rose Maylie is pretty much his defining characteristic.
The reason he isn’t living with his mother and Rose at the beginning of the novel is because he was staying with a rich uncle, who was planning out some kind of fancy political career for him, as a necessary preface to inheriting his fortune. But when Harry realizes that Rose can’t or won’t marry someone whose public life might expose her (and her questionable birth) to ridicule, Harry changes his entire career path. He ends up becoming a minister in a little country church, marrying Rose, making babies, and living happily ever after.
The only other time we see Harry is when he’s on horseback, egging on the crowd outside the house where Sikes is trying to escape. Harry takes an active interest in capturing Sikes and bringing him to justice—partly out of a sense of what is due to Nancy, and partly, we assume, out of affection for Oliver. So Harry’s not really one of the better-developed characters of the novel, but it’s easy to assume that, if you dug more below the surface, you’d just find more of the same. He’s not a complicated guy.