For much of the book, Snow isn't present, but his shadow (or maybe his nasty smell) looms over the characters and events. He's frightening even though he's not even around. In Chapter 1 alone, the trademark rose that he leaves in District 12 for Katniss to find in her old house is enough to make her freak out completely:
No one will fully understand – how it's not just a flower, not even just President Snow's flower, but a promise of revenge […] Positioned on my dresser, that white-as-snow rose is a personal message to me. It speaks of unfinished business. It whispers, I can find you. I can reach you. Perhaps I am watching you now. (1.49-50)
Snow has no problem authorizing cruelty and violence. For the majority of Mockingjay, he's the ultimate enemy. All Katniss can focus on is her desire to kill Snow – at some points, that's really all that keeps her going. She blames him as the force behind all the evil the districts have faced. He tortured Peeta and Johanna; he has people killed without a qualm. Finnick reveals that Snow is a pimped out Hunger Game victors and even resorted to murder in order to rise in the political ranks. Snow hasn't kept his seat as ruler through popular votes or good will: he's poisoned, scared away, or manipulated all his opponents.
Snow has a great deal of blood on his hands, which is somewhat ironic when you consider that his last name is Snow. A name like that implies whiteness, purity, and the cold. Snow is definitely cold, but he's about as far from pure as you can get. His impurities literally start leaking out as he ages, because he starts coughing up blood and smelling really foul. At one point in the text, Katniss finds out that's why Snow has roses everywhere: to hide his gross smell.
At the end of the book, when Snow and Katniss finally speak again, he's basically a broken man. He has lost and knows he's about to die. But he doesn't give up without trying to twist the knife in Katniss one more time:
Even in my deadened, drugged condition, this [Snow's comment about Prim's death] sends a stab of pain through me. Reminding me that there are no limits to his cruelty. And how he will go to his grave trying to destroy me. (25.37)
During this conversation, though, Snow's plants an important seed in Katniss's mind. His suggestion that Coin has been out to win power for herself all along and has fooled both Snow and Katniss is just too compelling to pass up. Katniss doesn't want to believe him, but she can't stop thinking about what he said. She and the readers might never know the full extent of Snow's evil, but one thing Snow does teach us is that evil isn't limited to just one person. And if you focus too much on one enemy, you might lose sight of the one standing right next to you.