Your question has to do with characterization, which is the art of creating a character. Characterization can be direct, in which the author simply explains elements of the character to the audience, or it can be indirect, in which the author describes things the character does, says, or feels as well as relationships with other characters or observations by other characters.
To create a character's desire, you could directly or indirectly characterize. Sometimes, the most effective method is in fact a combination of the two.
Direct Characterization: As the author, you could include within the narration an explanation of the character's desire, saying clearly whom or what the character desires.
Indirect Characterization: There are different methods you could use here, all of which are effective options.
- What the character says: A character's dialogue or inner thoughts could reveal desire
- What the character does: A character's actions, including the ways in which he/she says things, can imply certain feelings, including desire.
- A character's relationships: Whom a character is close to or, on the other hand, does not like, can reveal certain things about a character that could indirectly explain a desire.
- Other character's observations: What other characters see in a character can reveal desire, including their discussions about the character, observations of the character's behavior, conversations with the character, etc.