During the first book Snow seems to be your average President, although by the end of the book you begin to see his evil side.
Because he wasn't elected democratically, Snow understands that not everyone in the Capitol and the districts wants him as President, so has to threaten and kill people to keep his position. One of the people that hates him most is Katniss Everdeen, who, after winning the Hunger Games alongside her lover Peeta Mellark, is threatened by President Snow because he believes that she is a threat to Panem and that she must be contained. As Snow says in the end of the first book "A spark is fine, as long as it is contained". Here he is referring to Katniss as the spark, and is saying that Katniss is a treat to the Capitol, and that she must be controlled. Snow also visits Katniss' house in the beginning of Catching Fire to discuss the problem and inform her of how she has to fix it, and what will happen if she doesn't.
This problem first comes around when Katniss and Peeta attempt to make a fool of the Capitol by committing  double-suicide in the arena, and the Head Gamemaker stops them, allowing them both to be crowned victors. Snow is unimpressed by this move as he sees Katniss as  major threat to the Capitol, and has Seneca Crane killed because of it.

As the book series goes on, you begin to get more of a sense of Snow's evil intentions and just what he is willing to do to keep control of Panem.
The book begins shortly before the beginning of the Victory Tour, when Snow visits Katniss to discuss the consequences of attempting a double-suicide in the arena. He tells her that she has to convince the rest of Panem that her and Peeta are madly in love, and threatens to kill her closest friends and family if she fails. She fails to do this, and instead sparks a rebellion in Districts 8, 4 and 3. Snow becomes worried by this and plans to have Katniss and Peeta killed in the arena during Quarter Quell, in the hopes that this will bring the districts back into line. This is unsuccessful, and by the end of Catching Fire nearly all of the districts are openly rebelling including District 12, which Snow has bombed. This bombing kills Katniss' close friend Madge and 7,000 other people; nearly the entire population of District 12.

By the third book the country of Panem is in havoc, with Districts 1-11 openly rebelling. A large amount of this book takes place is District 13, where Katniss and the surviving victors are hidden. Because the Capitol cannot attack District 13 without the risk of being bombed, the victors are able to openly rebel against the Capitol without the risk of repercussions. One tribute who takes full advantage of this in Finnick Odair, spilling secrets about President Snow to Panem during an attempt to rescue Peeta. Finnick reveals that President Snow is a cunning, ruthless leader who is willing to do anything to keep his position as ruler of Panem. "Such a young man when he rose to power. Such a clever one to keep it. How you must ask yourself, did he do it? One word, that's all you really need to know. Poison". Finnick points out the deaths of numerous high-profile Capitol citizens who turned against Snow, stating that "Snow drinks from the poisoned cup himself to deflect suspicion."
At the end of the rebellion Snow is tried and convicted of murder, and is sentenced to death for his crimes.