When writing a literary analysis on Thomas Hardy's "The Melancholy Hussar of the German Legion," I would suggest that the most natural starting point is to think thematically. What are the key ideas or meanings which drive and/or shape this story, and what might Hardy be trying to express within it?

When thinking on this thematic level, a key hint might be found in the story's title, with its stress on the word melancholy. Note, however, that the Hussar, Matthäus Tina, is not the only character suffering within this story. In fact, the term melancholy can be applied to just about every major character in the story, with each of them feeling trapped or confined in their own way.

Matthäus himself has been forced into the army and taken miles away from home. Phyllis, the story's main protagonist, feels trapped by her confinement in her father's house and is left uncertain and in suspension about her fiancé's interest and feelings towards her. Meanwhile, her father himself is described as falling into isolation, and gripped by "the increasing perception that he had wasted his life" (Hardy, "The Melancholy Hussar of the German Legion," chapter 1), and finally, there is Humphrey Gould, and the revelations about his own married state in chapter 5.

Each of these characters is, in their own way, at odds with the societal expectations and/or constraints that have been placed upon them, and are left in a state of unfulfillment. The particularities vary from case to case (as does the resolution), but the term melancholy can be applied to each.

I think this would be a solid starting point, from which you can delve deeper into the story and its characterizations to formulate a working analysis.