The answer is given by Varys in Season 3 Episode 4 when was talking with Lady Olenna Tyrell:
Lord Varys: Littlefinger is not long for the capital. A confidant of mine has told me that when he goes, Sansa Stark goes with him.
Lady Olenna Tyrell: And why have you come to me with this matter?
Lord Varys: Littlefinger was born with no lands, no wealth, no armies. He has acquired the first two. How long before he has the
army? Perhaps you'll laugh, but I know him better than most, and
this is the truth. Littlefinger is one of the most dangerous men in
Westeros. If Robb Stark falls, Sansa Stark is the key to the North.
Lady Olenna Tyrell: And if Littlefinger marries her, he'll have the key in his pocket.
Lord Varys: Which seems such a shame. Why should a man with such a low reputation steal away such a lovely bride?
Lady Olenna Tyrell: You must despise him, you're working so hard to undermine him.
Lord Varys: Actually, I rather enjoy him. But he would see this country burn if he could be king of the ashes.
Littlefinger's end game is much more ambitious than most people realize.
Littlefinger was born the minor of the Lords, with just a few acres of lands on Fingers. However, he wants to rise up. But that's not plausible as Seven Kingdoms were dominated by following ancient Houses with big names and rich histories:
- Royal Dynasty: House Baratheon of King's Landing
- House Baratheon of Storm's End
- House Baratheon of Dragonstone
- House Tully of Riverrun
- House Stark of Winterfell
- House Lannister of Casterly Rock
- House Greyjoy of Pyke
- House Arryn of Vale
- House Martell of Dorne
- House Tyrell of Reach
He wants to be the King but how must he do that unless there is a power vacuum? He explains his philosophy to Varys:
Chaos isn't a pit. Chaos is a ladder. Many who try to climb it fail
and never get to try again. The fall breaks them. And some, are given
a chance to climb. They refuse, they cling to the realm or the gods or
love. Illusions. Only the ladder is real. The climb is all there is.
He wanted to create chaos which he would use as a ladder to climb up in social hierarchy. And of course he needed a Power Vacuum with waned power of the Great Houses, possibly extinction of some, so that their place could be filled by House Baelish. He does not care about the price, he would see the realm burn if he could be King of the ashes as Varys puts it.
So How did he plan to deal with the great Houses?
- House Baratheon of King's Landing: He was not too much bothered by it. He knew the secret behind Joffrey's, Tommen's and Myrcella's birth. He also knew that Jon Arryn and Stannis Baratheon suspected foul play and were on verge of actually finding out what was wrong. He let them do it, even facilitated them by Housing one of King Robert's whore and their bastard in his brothel. He wanted that secret to come out so that there could be a war of succession. When Robert was killed by Cersei (As he must be, since Cersei was getting more and more nervous each day and Robert was getting wilder every day), this House would go extinct de-facto and that will spark a war, If Petyr Baelish could help it.
- House Baratheon of Storm's End: He knew that Renly and Stannis could barely stomach each other so He was certain that it was unlikely that they would form an alliance to win the Iron Throne. He wanted them to battle hard and long while he could create other strands of the web.
- House Baratheon of Dragonstone: Stannis Baratheon suspected something about birth of the Princelings and he was very close to finding substantial evidence. When Jon Arryn died suddenly, that made Stannis believe that it must have been Cersei who did it to thwart their investigation. So Stannis bided his time until Robert left for Winterfell and then he ran back to Dragonstone and refused to return. He was sure now that they were right and Lannisters wanted to usurp the throne.
- House Tully of Riverrun: House Tully was a complex case. How to get them involved in the War of Succession in such a way that they choose a side which will help prolong the war and bleed away power from the rival houses. The answer was in their Stark alliance. Get House Stark involved, House Tully will join the fray.
- House Stark of Winterfell: This was the main piece. House Stark was loyal to King Robert and had enough power to defy the Lannisters. So in the planned war, they must be made to stand against the Lannisters, the other party in the dispute. Which is why Petyr Baelish asked Lysa to write a letter to her sister Catelyn and lie to her that Lannisters had killed Jon Arryn. This will make Eddard Stark dislike Tywin Lannister even more and force him to investigate the murder. There Petyr Baelish will lay breadcrumbs to follow so that Eddard Stark would come to the same conclusion as Jon Arryn and Stannis Baratheon. So he helped Eddard Stark find Barra. And coup de grace, he lied that the knife used for assassination of Bran belonged to Tyrion Lannister, further deepening the mistrust of Lannisters by Starks. This was the last straw and eventually resulted in arrest of Tyrion by Catelyn Stark, an event which began the War of the Five Kings.
- House Lannister of Casterly Rock: If everything went well with House Stark, House Lannister will be automatically at war with them.
- House Greyjoy of Pyke: Not that important and on the wrong side of Westeros, there wasn't much House Greyjoy could have done to further Petyr's ambitions. But still a great naval Power, House Greyjoy could make the war longer by engaging either Starks or Lannisters. In any case, there is nothing to suggest that Baelish had anything to do with their rise.
- House Arryn of Vale: If Jon Arryn died as per plan, Vale falls under regency for Young Lord Robin Arryn. And the regent was Lysa Tully who was being manipulated by Baelish for a long time. That means that Vale was already in Baelish's hands by proxy even before the war started.
- House Martell of Dorne: Unclear plans for House Martell if any existed.
- House Tyrell of Reach: As upjumped Stewards, Tyrells had achieved something what Petyr wanted to gain through a stroke of luck. Except that Petyr's plans were bigger.
So Petyr had to kill Jon Arryn because:
- That will make Stannis Baratheon sure that their suspicion was right and that he is the true heir.
- He could use Lysa to put the blame on Lannisters, creating a cold war between representatives of Starks and Lannisters in King's Landing.
- He could use the inquiry as a pretext to reveal to Eddard Stark that the Baratheon princelings were bastards born of incest thus sowing seeds of the war of succession.
- Get hold of the Vale in eventual war and then use that power to plot the next steps to the Iron Throne.
As it happened, War took an interesting turn and Sansa Stark was left as sole heiress of Winterfell. With Vale under his control, Petyr Baelish believed it would be more beneficial for him to marry the heiress to the North and thus use the combined power of Riverlands, North and unscathed Vale to break the crumbling great Houses in aftermath of the War.
Petyr Explained his final motives in Season 4 Episode 4:
Petyr 'Littlefinger' Baelish: So many men, they risk so little. They
spend their whole lives avoiding danger, and then they die. I'd risk
everything to get what I want.
Sansa Stark: And what do you want?
Petyr 'Littlefinger' Baelish: [pauses] Everything.
He made it clear to Sansa in Season 6 Episode 10:
Every time I am faced with a decision I close my eyes and see the same
picture. Whenever I consider an action I ask myself, 'Will this action
help to make that picture a reality'. I pull it out of my mind and
into the world, and I only act if the answer is yes. A picture of me
on the Iron Throne, and you by my side.
And of course about his actions he had following thing to say:
A man with no motive is a man no one suspects. Always keep your foes