What does the TARDIS think of The Doctor's companions?

It's difficult to say... Often the TARDIS doesn't seem to act so much like a person as it does a tamed animal like a horse or elephant. It's sentient, but as much a human being as the Doctor is, really. The Edge of Destruction showed it may have possessed a fondness for the Doctor and Barbara in particular as it uses the other two to try and warn everyone of the danger. The reason why it elected Susan as a medium was obvious in retrospect, i.e. her telepathic awareness, but it had a choice between either Ian and Barbara and it chose Chesterton. Maybe in part because Ms. Wright was the one who slowly came to accept the TARDIS's existence while Ian had rejected it outright.

You know, the TARDIS doesn't have as much of an antipathy towards Turlough as one might expect given that he tried to destroy her in Terminus. She certainly wasn't fond of Charley turning up before she should have and the relationship between the ship and Clara was... pretty bad. Bad enough that the TARDIS actually left the Doctor over it in Pay the Piper. It and Frobisher seemed to have a good enough relationship that they played chess together in Profits of Doom (even if Frobisher thumped the console with his foot when he lost), but even she had her limits for what she considered acceptable behaviour if The Holy Terror is anything to go by. Hell, he may have been the first and possibly only companion to make the ship go on strike, so I'd say it was a fairly abrasive acquaintanceship. Ace and the alternate universe TARDIS got on rather well, better than it did with the Seventh Doctor whom it didn't trust.