What is Littlefinger's overall goal

Why would Littlefinger want this person to be dead?

Littlefinger was one of the main drivers of the war, and the assassination of Jon Arryn was part of it.

Littlefinger had Lysa Arryn kill her husband Jon Arryn - Ned Stark's friend - who had investigated the lineage of the incested Lannister children.

Littlefinger involved Tyrion in the carrying of knives, and in the books he knew full well that it was indeed Joffrey to create more havoc. As mentioned above, Joffrey was the person who hired and armed the assassin in the books - and Littlefinger knew it. To impress Robert (who said it would be right to get Bran out of his misery), Joffrey stole the dagger and hired the assassin.

However, on the show, in the season seven finale, it is revealed that the knife was Littlefinger's own - Bran reveals that the knife is his at his trial, allowing Sansa to work out the implication that the enmity is between the Starks and the Lannisters Thanks to Littlefinger's forbearance.

These two threads work to pit the Lannisters and Starks against each other. Thus, Littlefinger can both have one of the two people who will prevent him from separating Sansa Stark from her family and bring himself one step closer to the throne after betraying Ned and perishing the events that led to the war in action.

Knowing this, Littlefinger's chat with Varus becomes a tacit admission of his hand to start the war by getting the balls rolling. He admits wanting the throne and contributing to Lysa Arryn's widowhood in a response as he talks about the throne:

Lord Varys: The Lysa Arryn of Chairs. It's a shame that you had to be satisfied with your second choice.

Petyr 'Littlefinger' Baelish: Early days my friend. It is really flattering that you are so afraid of the prospect that I will get what I want.

The assassination of Jon Arryn is one of the main reasons Ned Stark went to King's Landing. His suspicions of the Lannisters were only heightened by the attempt on Bran.

Littlefinger admits having stopped Sansa's marriage to Loras Tyrell:

Lord Varys: Thwarting you has never been my primary goal, I promise you. But if you don't like to see your friends fail now and then.

Petyr 'Littlefinger' Baelish: You are so right. For example, when I foiled your plan to give Sansa Stark to the Tyrells, it made me feel unmistakably delighted there, to be honest.

Sansa's captivity was a major cause of the war, and her marriage to the Tyrells was supposed to be her way out of Lannister's captivity.

Given our near-omniscient point of view, which Bran clearly has some insight into, we see that Littlefinger is absolutely to blame for starting the war - and on purpose, because the chaos is a ladder and he intended to climb the top - he wanted to be the king, and he planned a way to eliminate everyone in his path.