It's very hard to come up with why Littlefinger does anything, since he's very private with his true intentions. All we can come up with are a few guesses. As I see it, there are only two reasons Littlefinger might do anything: to advance his power and political position, and for pure personal reasons.
Littlefinger lost the hand of the love of his life because his name had no power behind it, and ever since he has been working hard to change that. In the span of 15 years or so, he went from the son of an impoverished lord to the Master of Coin of Westeros, Lord Paramount of the Trident, the Lord Protector of the Vale and Lord of Harrenhal. And it's quite obvious he doesn't have plans to stop.
Assassinating Joffrey accomplished several things:
- Removed Joffrey, who was by all accounts hard to control. In his place is now young Tommen, who has no real power of his own yet. Making the throne much more easier to influence.
- Get the favor of Tyrells who now wield great power in the council through Mace Tyrell, the Kingsguard through Loras Tyrell, and the throne itself through Queen Margaery.
- Gained control of Sansa Stark, who is (as far as everyone knows) the heir to the North. Now if Littlefinger's scheme of marrying Harry the Heir to Sansa works, he will gain control of the North and the Vale.
So through all these machinations Littlefinger can potentially control the North, the Vale and the Riverlands, as well as having considerable influence with the Tyrells who control the South and are largely the power behind the Iron Throne (now that Tywin and Cersie have been neutralized). So one can argue that Littlefinger is now, or is going to be, one of the most powerful men in Westeros if not the most powerful.
Here it is pure speculation, since we have no insight on how Littlefinger thinks. I can't think of Littlefinger having any daggers to hone with Joffrey. However, we do know he still retained his love for Cat, and he thinks she was killed by the Lannisters.
More likely, I think, is that Littlefinger harbors a grudge with all the Great Houses. After all, he lost Cat to the heir of House Stark. But even with House Stark all but extinguished, his machinations continue. Which hints that his revenge plot (if it does indeed exist) encompasses all the families that have looked down on him in the past.