Haymitch Abernathy is a drunk.
He spends most of the movie inebriated, and when people take his booze away from him, he gets downright violent. Yet he's Peeta and Effie's advisor, being the only other person from District 12 who ever won the Hunger Games. As much as they don't like him, they have to trust him.
Why? In addition to being a survivor, he's a celebrity to the people in the Capitol. He can lobby the rich and powerful to send goodies to Peeta and Katniss. That can be the difference between life and death in the arena, and having been through it himself, he knows how important those goodies can be.
Initially, Haymitch looks like an alcoholic nitwit who cares only for the next bottle he's about to guzzle down:
KATNISS: So, why are you here, then?
HAYMITCH: Oh. The refreshments.
No mention of, you know, helping the kids who are about to go to slaughter? Not a great look, Haymitch. Having said that, we're inclined to cut old Haymitch a little slack. After all, he's been doing this ever since he won the Hunger Games himself, and every year he gets to meet two nice young people who subsequently die horribly within a few days.
Every. Single. Year.
That can wear on a guy, but because he's obligated to do this, he has to deal with it over and over again. So yeah, he drinks. We can't condone it, but we can at least see where he's coming from.
This time, though, Haymitch sees a glimmer of a chance in these two. (Or maybe he does it with every District 12 Tribute, we may never know.) He pushes them harder to earn sponsors, to behave in ways that draw their attention, and to otherwise do everything they can to make it through the Games.
He knows how to do this because he's played the game for years. Appearance is everything in this world, and being memorable is often more important than being tough. Look, for example, at the advice he gives Katniss on impressing the judges:
HAYMITCH: I don't know how else to put this: Make sure they remember you.
Then later, when Peeta confesses his love for Katniss on live TV, Haymitch jumps on it even though Katniss doesn't like the idea at all. Why?
HAYMITCH: Now I can sell the star-crossed lovers from District 12.
KATNISS: We are NOT star-crossed lovers.
HAYMITCH: It's a television show, and being in love with that boy might just get you sponsors which could save your damn life.
He's actually talking about power here. Katniss and Peeta are victims of this system: sent off against their will to battle to the death against Captain Steroids and the other killbots from Districts 1 and 2. As such, there's not a lot they can do about their situation. But if they can stand out on television, they become celebrities, and fame can get them the things they need.
No one knows how to play this game better than Haymitch, and he works overtime at it. It's a sick game, but he can play it with the best of them. As a result, Katniss gets some goodies that end up saving her life. While Haymitch may be an unbelievable pain in the butt, he's also perhaps their biggest ally in the Capitol.
That said…seriously dude, stick to iced tea.