Hypothesis:

Quirrellmort intends to upload his mind into Harry's body soon, as soon as Harry is Dark enough. Voldemort will become the Boy-Who-Lived. And Quirrellmort wants or needs this to happen within the next few months.

Evidence:

If Quirrellmort were only after the Philosopher's Stone and training Harry for a long career, he'd keep his own cover intact as long as he could. Instead, over the last few story months, Quirrellmort has cheerfully all but ruined his cover in favor of giving Harry chances to turn Dark.

• Quirrellmort got the Dementor brought to Hogwarts, waited until the last moment to observe Harry's wand by the Dementor's cage, gave wrong advice about how to help Harry recover from the Dementor-induced personality change, and persuaded the other wizards to let Harry face the Dementor again.
• Quirrellmort took Harry to Azkaban soon after seeing Patronus 2.0, leading to more Dementor contact and the recovery of Bellatrix, Quirrellmort's preferred assistant for critical tasks (like, say, ritual magic to download yourself into your Horcrux's body).
• Quirrellmort (as H&C) set up Hermione for Draco's attempted murder, thus both cutting off Harry from the person who's his best influence against being Dark, and motivating Harry to embrace his Dark side more in order to rescue or avenge Hermione.

The first stunt made Dumbledore suspect a plot, the second showed that Voldemort had returned, the third that Voldemort was in Hogwarts. But it's all been worth it to Quirrellmort to hurry up Dark!Harry. Why?

Perhaps because Quirrellmort is running out of time in his current body.

• On the day he killed Rita Skeeter, she observed Quirrellmort had his hair falling out.
• Harry has noticed Quirrell looking visibly older.
• The curse on the Defense against the Dark Arts position demands a terrible conclusion to his year - or at least the appearance of one.

And if Quirrellmort doesn't intend to be around as Quirrell much longer, what does he intend? We have clues.

• We know from the prophecy that Quirrellmort and Harry can destroy each other's spirits.
• Quirrellmort has said that Harry's sense of doom between them is precisely of Harry's doom.
• We suspect from canon combined with "Dark Harry" moments that Harry is an accidental Horcrux containing a partial copy of Voldemort's mental circuitry, and possibly some of his power too.
• Quirrellmort suggested a plan where Harry would be seen to fight the returned Voldemort and defeat him.

The obvious answer is that Quirrellmort intends just the plan he told Harry: Harry will indeed be seen to fight Voldemort and "defeat" him. And "Quirrell" will die, probably having been revealed to be Voldemort. But Quirrellmort... will have downloaded himself into Harry's mind, and so will win the duel he seems to lose.

Just as before, a single clash of spells between Voldemort and Harry Potter will lead to the destruction of "Voldemort"(Quirrell.) And "Harry" will walk away triumphant. But "Harry" won't be Harry any more.

If Quirrellmort's plan succeeds, Harry as we know him will cease to exist. Voldemort will go on in triumph -- as Harry Potter, the boy who destroyed Voldemort twice over. Harry will be the beloved hero of magical Britain -- and Voldemort inside.

I suggest Quirrellmort's top priority is to turn Harry fully Dark before the end of the year, so he can safely download into Harry.