What are some ideas to help me with writing "The Ugly Duckling" in the voice of the Ugly Duckling?

When you write a short story, consider it as three parts, called the story sequence. Part I introduces the characters and the setting. In this part you show the main character in his or her normal setting and describe what the characters are like in the beginning of the story. For this story, you would describe where the ducks live, what they do on a normal day, and introduce the Ugly Duckling and, perhaps, name a couple of other ducks as characters.
Part II of your story is the Problem/Conflict section. Here you want to elaborate on the main character's want or need. Begin by having something bad happen to the Ugly Duckling, an incident of teasing. Describe from the Ugly Duckling's perspective how the teasing makes him feel and what he wishes his life was like, instead. You may have a couple of increasingly unkind instances of teasing in this section.
Part III of the story is the Climax/Resolution section. Here the rising action of the story, the increasing cruelty of the other ducks, reaches a peak. Something pushes the main character over the edge, so he has to make a decision that sets up the resolution of the conflict. In this story, he flies away in the spring and joins a group of beautiful birds, thinking he will be killed by them because of how ugly he is. But they welcome him. When he sees his reflection, he sees he is one of them. That is the resolution. He lives happily ever after, so describe how he feels upon seeing his own beauty and being part of a group who accepts him.
Writing from the Ugly Duckling's perspective doesn't mean you have to tell the story in first person (using "I" and "me"), but that is probably the easiest way to do it. Just keep in mind the three-part format of the story as you write: Setting, Conflict, and Climax/Resolution.