(Spoiler alert for, well, Game of Thrones TV show up to Season 5 Episode 6, obviously.)
This question is about the events in the show, not the books (in which things happen very differently) - although answers based on book information not presented in the show, and which does not contradict what we know from the show, are acceptable :-)
So after the Vale affair, Petyr Baelish offers Sansa to the Boltons as a wife, despite saying he expects Stannis to beat them and take Winterfell.
That in itself does not make much sense, but his motivation seems to be revealed in this episode: He can make the Boltons look like traitors to Cersei. Truth be told, that's rather lame, since if Qyburn is doing any sort of work as Master of Whispers, he should have informed Cersei that it was Lord Baelish who offered Sansa, pulling the rug out from under his argument at the tete-a-tete with the Queen Mother.
But, let's put this contrivance aside and assume he had Sansa marry Ramsey Bolton for this purpose.
Well, what does this get him, exactly? I mean,
Cersei does not give him troops, nor money, nor does she get Lannister allies to act in any way. He's been promised to get to be 'Warden of the North' if he wipes out Stannis. Is he really in a position to lead the Vale to war against Stannis in support of the Lannisters? Even with a royal decree, I don't see how that's happening. Maybe he can manipulate lords who owe him money or something; but even then - it's not just Stannis, it's Stannis and the Iron Bank. Seriously, I don't see that happening. Even if he had pulled this off - who said he would win? Petyr Baelish never led men into battle in his life, and now he's to carry out a foreign invasion against a seasoned and accomplished military commander? Who will likely enjoy some support from the surrounding population? Finally, Baelish has a fixation on Sansa, why would he put her head on a spear for Cersei, especially having invested so much effort in getting her out of King's Landing?
I must be missing something. How are all these Sansa-centered machinations of any utility to Lord Baelish?
Edit following Episode 7:
So now Baelish and Tyrell have sprung the trap on Cersei. This makes the request of sending armed men up north even more ridiculous, since even if he'd gotten her to accept that, she wouldn't get around to arranging it. Also, it's not even clear whether or not he's going to have that royal decree of his. This leads one to believe that everything he told Cersei was essentially babble to keep her distracted. That's fine, i.e. makes Baelish seem less of an idiot. But then, the question remains: What's the use of sending Sansa over to marry Ramsey Bolton?
Edit following Season 5 Finale:
Looks like whatever it is Baelish was planning, it isn't quite working out. He should probably have kept her in the vale - unless he has some fiendishly-clever idea I can't even fathom.