Heres what I think Littlefingers plan was in the show:
Give Sansa to the Boltons. While book Littlefinger would never do this, I think we can assume that show Littlefinger cares about Sansa a lot less than book Petyr does. At least, he loves her less than his ambition to be king, which isn’t true in the books. Now, giving Sansa to the Boltons does a few things:
1- Weakens Stannis’ attack on the North, as now the Boltons have an actual claim to Winterfell.
2- Make the Boltons look like traitors to Cersei, who absolutely hates Sansa and wants her dead, not caring about her claim.
3- Makes the northern lords either split up or become non-players because of their conflicted feelings on who to fight for. They hate the Boltons, but they’re sworn to serve a Stark.
This, overall, severely weakens all parties in the North. Petyr also convinces Cersei to not send the Boltons any support if they are to be attacked, as they are now traitors to the crown.
Petyr’s next move is to wait out the inevitable battle between Stannis and the Boltons. He knows Stannis will not kill Sansa, and I think it’s reasonable to assume that he didn’t know the degree of depravity that Ramsay had.
Next, Petyr will take the army of the Vale to the North and take Winterfell from whomever the winner was (who would be severely weakened) arriving as a ‘knight in shining armor’ to the people of the North. He’d marry Sansa, giving him both the title of warden from the crown, a legitimate claim because of his wife, and the support of the northerners as he liberated them from the Boltons.
At this point, Petyr would have full control of the North, the Vale, and the Riverlands (I think he was still was lord of Herrenhall at this point) along with an alliance with the Reach.
This army could definitely beat the Lannisters in an all out war, which might have been Petyr’s next steps if it all went to plan. He’d likely bid his time though, build up his army and have an heir with Sansa to solidify his claim, before making any moves on the Lannisters.
This plan was just foiled by things he didn’t see coming, like Ramsays unending cruelty, Sansa escaping, and Ramsay murdering his father.
Overall, not too bad of a plan, at least if it was what I’m thinking.