Haymitch is a prior winner of the Hunger Games; he was forced to kill friends and innocents to survive, and he doesn't want to participate in the charade put on by the Capitol. To keep himself from thinking too hard about the children who are forced to kill each other, he drinks heavily. It is only after he becomes aware of the will of both Katniss and Peeta to survive that he starts to take the possibility of their survival seriously. Haymitch hates the Capitol and their oppression, but knows that rebellion is a useless gesture. He also shows "survivor's guilt," feeling deeply depressed about his survival over others. As the books progress, he begins to feel hopeful about the possibility of overturning the Capitol, and feels a fatherly concern for Katniss.
Haymitch is completely detirminded to help Katniss survive. He knows that she has a better chance on the outcome and she is strong enough to be an influence to the rebelion.