I'm not aware that Elizabeth hated Lettice before Lettice's relationship with Robert Dudley. Lettice's mother, Catherine, was Elizabeth's cousin and was very close to the Queen serving her a Chief Lady of the Bedchamber. Lettice also served Elizabeth as a Maid of the Privy Chamber and I'm sure Elizabeth would not have given her that position if she hated her. Diego Guzmán de Silva, the Spanish ambassador, described Lettice as one of the Queen's favourites in 1565 so she was still in favour then, when she was married to Walter Devereux. However, in that same year she did flirt with Dudley and this made the Queen very jealous and angry.
Lettice secretly married Dudley on 21st September 1578 and it was this act which led to Elizabeth banishing Lettice from court and labelling her the "she-wolf". She never forgave Lettice and they were never reconciled.