If Oz were high school, the Scarecrow would be the nerd, the Lion would be the jock, and the Tin Woodman would be guy who wears black lipstick and listens to the Cure. The poor guy is convinced he doesn't have a heart, but the amount of time he spends emoting all over the group suggests otherwise.
To be fair, the Tin Woodman's history is super dark and morbid, so it's hardly any wonder he's confused. His backstory is easily the most disturbing moment in the book. Long story short, he was once a human whose axe was enchanted by an evil witch. That caused him to chop off all of his own limbs, one by one. A local metal worker was able to rebuild him out of tin, but the whole experience left him so disgruntled he lost the love of his life and has been living by himself in the woods ever since.
To make matters worse, a year before Dorothy and the Scarecrow found him in the woods, he got caught in a rainstorm and froze (or, technically, rusted) into place. In that year, he had a lot of time to think on love and loss, and now the Tin Woodman wants—nay, needs—a heart. He's tired of being a recluse. He wants to love again.
On the road, it quickly becomes evident to everyone but the Tin Woodman himself that he isn't heartless at all. You might even say he cares too much. "Once, indeed, the Tin Woodman stepped upon a beetle that was crawling along the road, and killed the poor little thing. This made the Tin Woodman very unhappy, for he was always careful not to hurt any living creature, and as he walked along he wept several tears of sorrow and regret" (6.44). He also cries when the Scarecrow gets separated from the group, when the Wizard leaves in his hot-air balloon, and when Dorothy goes back to Kansas, to name just three occasions. What a sweetheart.
The Tin Woodman is finally able to enjoy the heart he's always had when the Wizard gives him a fake (if lovely) one made out of silk and sawdust. Once he's found himself (and safely seen Dorothy off to Kansas), he declares himself ready to start a new life as leader of the Winkies, who he'll no doubt lead with both his head and his heart.