In an attempt to find Quirrell's motives, I have listed the evidence I have about him, and now have a theory I have not seen on LessWrong or anywhere. I did it mostly mentally, but I'll try to put down all the evidence I took into account as unbiasedly as possible. I assume you know Quirrell = Voldemorte = Tom Riddle.

-Quirrell said Harry's wish was impossible. The wish was that Quirrell come back again the next year as the Defense Professor at Hogwarts. He also burned the paper on which the wish was written and he did not tell the audience what it was. If Quirrell intended to come back the next year, he would not have minded the question. Therefore, he is probably only staying a year, but he does not want most people to know.

-He took Harry's "Dark Side" explanation in stride, which normally should have sounded like a terrible excuse. He therefore probably knows something about it.

-Hermione claimed that Quirrell wanted the Dementor to eat Harry. I see no reason for Hermione to have lied. Quirrell stopped the Dementor's feeding after Harry had already turned to his Dark Side by telling Flitwick to remove the wand. After this event, Harry's sense of Doom grew stronger. I doubt that Harry's Dark Side coming out would have been a permanent change, however, simply He didn't seem to do much else to turn Harry dark, except making him say that his destroying of Skeeter was a good thing, and that can easily be explained by him being annoyed by Harry's whining. I take it from this that he wanted to get Harry's dark side out more, and that that was the intention behind the Dementor, but he's not trying to turn Harry evil per se. -Quirrell also seemed very eager to believe Harry would have the ambition of becoming a Dark Wizard. I take it from this that he expected Harry's Dark Side to have more control over him.

-He gave a speech stating that magical Britain should unite with a mark under a good leader, inferring that it should be Harry, and claimed that Harry's next speech which said the opposite was interfering with his plots. He also has helped the students of Hogwarts grow stronger and wiser. I therefore presume that he wants light strong, because he could just as easily have been a bad professor, given Hogwart's track record.

-He was able to go to Harry's house thanks to a phone book, which probably required Harry's Stepfather's last name to find. This means he couldn't have kidnapped Harry before meeting him in Hogwarts, but that he could if Harry went back home for the summer.

-He rescued Bellatrix Black from Azkaban, but tried to do so subtly, and it would probably have been impossible to save her without Harry learning the True Patronus, which was not predictable. I know 2 possible uses to Bellatrix: come back to life, or get his wand back. Quirrell had the opportunity to kidnap Harry and still have Bellatrix Black (right after they had saved her, he could have just not gone back to Mary's Place and taken Harry with him), which would have allowed him to come back to life in full strength, but he didn't take it. The plan was supposed to go unnoticed, but that failed, which made Dumbledore, Minerva, and Snape suspect Voldemorte is coming back, preventing Harry from going back home for the summer. He then wanted to make a plot that would make everyone believe Harry had defeated Voldemorte again, so I presume he doesn't plan on coming back as Voldemorte, at least not yet.

-Quirrell had an opportunity to kill Dumbledore (when he was weakened by the Dementor), but he didn't take it. His plan is therefore not to kill Dumbledore, or at least he needs to get other things done before killing him, and taking that opportunity would have prevented him from accomplishing them because he would have been ousted from Hogwarts.

-Quirrell knows how to get back to life by stealing a body, though it has negative side-effects that get worse over time.

-Harry and Quirrell are very similar in terms of wand, parseltongue, and they have some weird magical resonance, and a sense of doom when they approach each other.

-Quirrell has told Harry not to trust Dumbledore. After saying "I don't suppose you have a common enemy" to Harry, Hermione, and Draco, he glanced at Dumbledore almost imperceptibly fast. The only thing I can make of this is that he doesn't like Dumbledore.

-Quirrell blocked attempts to identify him by the aurors who knew he was not Quirrell, but the Hogwarts map couldn't find him when asked to find Tom Riddle, his true name. I'm thinking body-stealing makes you that person for all things linked to the name of the purpose, which would make the spell identify him as Quirrell. He doesn't want the aurors to identify him as such, however, because they already know that he's not Quirrell, unlike Dumbledore. Update: as moritz has pointed out, Quirrell wasn't there when Dumbledore asked the map to find Tom Riddle. I also assumed that Amelia Bones was referring to Tom Riddle when she was talking about a person sorted into slytherin in his same year, though Eliezer has confirmed that this is not so by changing the year of birth. So I have no evidence for this anymore. But come to think of it, even if the map would identify Quirrell as Tom Riddle, that probably means that Quirrell doesn't know about the map, and therefore he hasn't planned for it.

-Quirrell tried to get rid of Hermione. I don't think he intended to get rid of Draco, because he saved Draco from death, and had Harry not beaten all odds and saved Hermione, Draco would probably not have been removed from Hogwarts by his father. He probably expected her to go to Azkaban, but he would settle with her going to some far-away school. I can think of 2 motives for this: turn Harry to his Dark Side more, or get rid of Hermione (probably both).

So what is Quirrell plotting? I think he's planning on stealing Harry's body like he stole Quirrell's, at the end of the year when Harry goes back for the summer. He's been bringing out Harry's Dark Side, which he probably created inside Harry (maybe as a "horcrux", whatever that is), because that would allow him to better fit inside the body. He doesn't want to turn Harry evil, however; he wants to become the hero again as Harry Potter, and become the ruler of Britain under a light mark. He's also trying to harm Dumbledore, though, because he doesn't like him and is a threat. Maybe he has other motives as well for harming Dumbledore. He got rid of Hermione because she is the most likely to notice a change in Harry, and he didn't do it during the Christmas vacations because it would be suspicious if both Harry changed and the Defense Professor disappeared at the same time, unless it was summer and the professor disappeared because he left for normal reasons.

Update 1: I read some more comments, and I've noticed that some people think Quirrell crippled Harry by turning him into an Occlumens and teaching him to lose. I disagree. Harry learned to pretend to lose, not to give up. I think this was more a measure to prevent Harry from self-destructing himself like he almost did when fighting Snape. And Harry becoming Occlumens, while preventing him from testifying under veritasaerum, was probably mostly to allow Quirrell to use Harry in his plots without the danger of Dumbledore legilimensing something important.

Update 2: It's interesting that I used an euphemism for the word "hate" when describing Quirrell's relationship with Dumbledore. The word "hate" just felt too strong to describe any emotion Quirrell could have towards others. Quirrell is far too good a manipulator, he even makes me feel as though he's too collected and calm to actually hate anyone.