how does the moment when Holden thinks about ducks characterize him?

In one sense, the ducks might symbolize resurrection – they always return in the spring. But you might have some difficulty arguing this, this Holden is more concerned about the ducks dying than he is excited as the prospect of them returning, and also, it's December, so spring isn't really on anyone's mind right now.

More likely, we can work the mortality angle here to explain Holden's obsession. You know the scene where Holden goes to the lagoon and looks for the ducks, and then starts thinking about his own death, and then ruminates on Allie? Exactly. That means all these things are related.