Well, let us look at the definition of psychoanalytical theory (now often called "psychoanalytic theory" as the more modern term) and then determine how this definition fits with Freud's case study of Dora. The Oxford English Dictionary defines Psychoanalytic Theory as follows:
A therapeutic method ... for treating mental disorders by investigating the interaction of conscious and unconscious elements in the patient's mind and bringing repressed fears and conflicts into the conscious mind, using techniques such as dream interpretation and free association.
In other words, Freud evaluates both the conscious mind and the unconscious mind through analyzing dreams or association or augmentation of fears that have been muted or repressed. Freud's "work on Dora" which is actually a case study, is really called Fragments of an Analysis of a Case of Hysteria. Dora's original name, of course, was Ida Bauer.
We can first see "Dora" as an example of Freud's Psychoanalytic Theory due to his investigation of Dora's unconscious mind and conscious mind. Dora was pretty quick in revealing her conscious mind to Freud. Turns out that her dad had a relationship with another woman (other than Dora's mom), and as a result, Dora felt that she was being give to the husband in that relationship as a pawn. The incident that began the therapy is that Dora slapped Herr K (the man in question) when he came on to her. It was Freud's interpretation of Dora's unconscious mind that caused such a problem for Dora that she stopped her treatment. Freud insisted that, unconsciously, Dora had a deep sexual attraction towards Herr K and actually desired to have the relationship with him. Due to Dora's quick exit, Freud had to declare the case a failure.
Next, we can see "Dora" as an example of Freud's Psychoanalytic Theory due to her repressed fears that Freud analyzes as well as the interpretation of her dreams. Dora had two main dreams interpreted. The first was Dora's house that was up in flames that the family escaped from (after they decided not to save Dora's mother's jewelry box).
Dora also had a second dream that Freud though revealed Dora's repressed fears. The second dream was more detailed about moving around in an unknown town and then her own home where she read a letter from her mother about both leaving home and about Dora's dad being dead. Dora tries to get to the station to attend the funeral, but is unable to travel there until the end of the dream when Dora finds that everyone is already at the cemetery. Freud compares both the jewelry box and the train station to Dora's virginity. These two dreams, of course, brought up Dora's repressed fears of submitting to Herr K.
Thus, as you can see, by analyzing Dora's conscious mind (through her speech) and her unconscious mind (through Dora's dreams) Freud uses his his Psychoanalytic Theory in order to diagnose and help Ida Bauer who had the name of "Dora" in Freud's writings to preserve her anonymity.