While it is true that Aunt Alexandra tried to get Calpurnia fired in a failed Finch family power play, Atticus's sister doesn't seem to hold any specific personal animosity toward Calpurnia. Aunt Alexandra arrived in Maycomb expecting to take control of the Finch household, and Atticus gave her free rein until she brought up the subject of dismissing Cal. Alexandra probably never expected her "soft-hearted" and non-confrontational younger brother to stand up to her, but Atticus put his foot down when Cal's future in the Finch family was questioned.
"... I couldn't have got along without her all these years. She's a faithful member of this family... and another thing, the children love her." (Atticus, Chapter 14)
Unlike Alexandra, who Atticus and his children only saw each year at Christmas, Calpurnia was both loved by the children and an excellent example of a motherly figure. While Alexandra spoiled her grandson, Francis, and her own son had little to do with her, Atticus knew that Cal had
"... been harder on them [Jem and Scout] in some ways than a mother would have been... she's never let them get away with anything..." (Atticus, Chapter 14)
Atticus treated his housekeeper as an equal, but Alexandra--who was not as color-blind to the races as her brother--saw Cal only as a Negro servant. Alexandra herself had employed a black man as her chauffeur (that's how she arrived in Maycomb), and she probably never envisioned that any outsider--and especially an African American woman--could come to be as important and beloved to the Finches as Cal. Alexandra begrudgingly accepted Atticus's excuse that he didn't want her "working her head off" during her stay in Maycomb, and she eventually accepted her position as the female head of the house, superceding Cal but never Atticus.