Define the character's wants. In writing a story, it's extremely important to know what motivates a character, because this is what will drive your plot. But it will also create a way for you to work subtext into the character's dialogue. What he actually means and feels come directly from these wants, and when you place the character in any given situation--whether or not that situation deals directly with the character's wants,--you can insert subtextual meaning into his dialogue.
Create a scene or an event that sets the groundwork for subtextual meaning. Subtext means that, on the surface, the scene can appear to mean one thing, when it actually means something else entirely. To create subtext in dialogue, you must determine what the scene is really about. For instance, a woman who is moving in with her boyfriend is having doubts about this important step in their relationship. These doubts are brought to the surface, but are within the subtext of the scene. Therefore, what the woman says during this scene will refer to the subtextual meaning.
Create a conversation between the characters. Get them talking. The discussion in the scene should cover what the action is ostensibly about. For instance, if the woman is moving into her boyfriend's apartment, create a conversation between the two in which they discuss just that. They can discuss where they will put away the girlfriend's belongings, how much closet space she'll need for her clothes, or whether the boyfriend should get rid of his own things to make room for hers. While on the surface, these discussions are normal given the situation, they can also provide the appropriate occasion for subtext.
Suggest something entirely different from what your characters are saying. In the example above, the woman and her boyfriend might discuss closet space. The woman remarks how small the space is, and wonders if there will be enough room for her clothes. The boyfriend tries to convince her that there will be, but the woman is still not convinced. "There's not enough space," she says. "Actually, this whole apartment suddenly seems smaller." The boyfriend responds, "My place hasn't changed. It's no bigger or smaller than the last time you came over." "I know," says the woman. "That's what's so odd about it." Here, the woman doesn't directly state her doubts about the relationship, but it is reflected in her observations about the apartment.
Use a symbol or metaphor in your story or scene, and have characters talk about it. Since the symbol will reflect the deeper meaning of the story, your characters' reactions to it will be reflected in the subtext of their dialogue. The apartment is one example: It's a metaphor for the woman's relationship with her boyfriend. It represents the serious turn this relationship has taken, and the woman's reactions to it will reflect her own changing feelings about it. When she states that the apartment has become smaller, she is actually referring to her own feelings about her boyfriend.