Rose-the-Cat (as it's her name not a title)
Talk about it here.
Rose would often try to help the Doctor cope with the loss of his former love, through various means. Most to all of these failed. Despite this, her time on the TARDIS did note a sharp contrast in the performance and character of the Doctor. While he would initially avoid leaving the TARDIS or having adventures at all, by the time she left he was not only actively seeking out trips once again; but he also eventually decided it was time for him to take on another companion.
Rose eventually left the TARDIS team, searching the cosmos for a new person to comfort as the Doctor decided it was time to find a new companion. After an immeasurable amount of time in stasis, Rose emerged to find the real Rose Tyler standing in front of her pod in a state of confusion.
The Doctor adopted Rose during a fit of depression, and named her after his now-lost companion Rose Tyler.
Owning a cat was initially far from what the Doctor expected it to be. Directly after bringing her into the TARDIS, Rose scratched the Doctor up and ran off. The Doctor would not see her for five days. When he did see her again, he lectured her for being "a bit rubbish," but Rose mainly ignored him during the conversation.
The Doctor at one point spent an entire day sulking in bed and crying over the loss of Tyler. Attempting to cheer the Doctor up, Rose-the-Cat managed to pilot the TARDIS to various locations in space-and-time to track down animals to hunt. She then presented their carcasses as gifts to the Doctor, which only made him more sad over the death of all of the various creatures. Rose did not comprehend why the Doctor was not grateful for her hard work.
The Doctor in another instance entered his closet to discover that Rose-the-Cat had pulled down all of his suits and sat on them, leaving them covered in neon hair. The Doctor planned to travel to Deva Loka to relax, but discovered that Rose had scratched up his long brown coat, ruining it. Enraged, the Doctor left Rose dangling outside in a space-suit as he considered how he was nowhere near as merciful as he used to be.
Rose eventually began to warm up to the Doctor's presence and accepted him as her owner. Rose at one point asked the Doctor if he had ever had any other pets in the TARDIS, to which the Doctor admitted that there was someone that Rose needed to meet. That "someone" was a personal version of K9 which the Doctor kept on the ship. Standard for most dogs and cats, the pair did not get along. Their first encounter ended with Rose attacking K9 with the Sonic Screwdriver and K9 entering "defense mode." The Doctor once again resulted to leaving the two dangling outside the ship as punishment.
While walking together, Rose-the-Cat suggested that the best option for getting over Rose-the-Human was for the Doctor to get a rebound companion. The Doctor was initially reluctant, but eventually agreed. The night of his "blind date," he stormed back into the TARDIS exclaiming that he had been stood up. In fact, his potential companion (a member of the Sisters of Plentitude) was in the TARDIS petting Rose-the-cat. Upon seeing this, the Doctor angrily reminded Rose-the-Cat that they were there to find a new companion for him. Afterwards, the Sister asked if she needed to leave.
Rose at some point was infected with fleas, and while the Doctor was initially playful in teasing Rose's newfound friendships, he eventually realized that it was becoming a serious health hazard for the feline. Refusing to use regular flea cures (which would "upset the natural balance") the Doctor took Rose to a planet made entirely of candy floss surrounded by a gas that was irresistible to insects. The Doctor placed Rose in the stuff and successfully cured her of her pestilence, but doing this also shaved most of the hair off of her lower body. In a "genius" solution, the Doctor put Rose in a miniature blue suit much like his own. While the Doctor and K9 were amused by the situation, Rose was not.
Rose and the Doctor eventually parted ways, with the Doctor deciding that it was finally time to find a new companion. Rose joined an "agency" which would send her through the universe looking for another heartbroken soul to comfort. After travelling for an immeasurable amount of time in stasis, Rose emerged from her pod to see her namesake standing directly in front of her. Amused, Rose-the-cat asked Rose-the-human if she was a fan of Chums. (COMIC: A Rose by Any Other Name)
Rose-the-Cat was, by no one's standards, human. Rose would often have trouble understanding the Doctor; partially because of his insistence on never moving on from his past, but also mainly because of basic misunderstandings which fell from her being a cat and him being a Time Lord.
Rose would often only think of herself when making decisions. While she often encouraged the Doctor to move on and find a new companion, it was more often than not because she found his constant ranting and bickering to be annoying. She was easily bored and easily annoyed, and she would never try to hide how she felt.
Rose showed a surprising lack of concern for the effects of time travel, and she would often cross her own timeline simply to avoid mildly annoying circumstances or events. She at one point sent a shockwave of paradoxes across all of time-and-space in an attempt to create a cat flap in the TARDIS' wall.
While Rose often would approach the Doctor with plans for dealing with his lingering emotions for Rose-the-Human, the plans would often be things that Rose simply wanted to do anyways. One one occasion, Rose hijacked the Doctor's blind meeting with a potential companion for the sake of attention. In other circumstance, Rose suggested the Doctor start a band to work out his feelings. The mission was a success, but Rose still attempted to revert the band ever being created simply because it annoyed her. (COMIC: A Rose by Any Other Name)