Why did Eisenheim make Prince Leopold feel guilty of murder?

Eisenheim fell in love with sophie when they were just kids, but since Sophie was royalty and Eisenheim was not, they were forbidden to see each other. Later in life, Eisenheim reappears in town and once he sees sophie again, plots to free her from her current situation so that the two of them can live together. He knows she still loves him because she is wearing the locket he made her.

The best way to do this without anyone getting suspicious is to make everyone think that sophie is dead. Making the emperor believe that he had killed her made it easy to execute the plan.

As far as a rivalry, the emperor is known to abuse women and once Eisenheim embarrasses the emperor during his private show, Eisenheim is banned from performing. The emperor is not a nice guy and it is portrayed that the country would be better off without him.

So, to answer your question, Eisenheim tricked everyone, including the emperor, into thinking that he had killed sophie to prevent him from trying to find her once she left (or trying to kill her when she tried to leave). But in order for the two of them to be left alone, the emperor needs to be either arrested, or dead. The emperor did not kill himself out of guilt, however, he killed himself so he would not be arrested after his plot of overthrow his father (The real emperor) was exposed.

There was no guilt and he didn't think he killed her. He committed suicide because of the charges (of killing her & mainly the charges of trying to seize the throne).

Here are a few pleas he said, trying to convince Uhl he had nothing to do with killing her and all the evidence were faked.

You fool. He's tricking you.

He's manipulating you. He had the sword at a command performance.

He could have pried the stones loose then and planted them here.

He has planted everything.

It's all a trick.It's an illusion.

And just before he kills himself

You're all fools.

Eisenheim wanted Sophie. They were in love, and she didn't want to marry Prince Leopold. Sophie had to be believed to be dead for her to be with Eisenheim. Framing the Prince was the perfect way to distract everyone from what was really going on.

As for why... The movie implied that the Prince had assaulted one of his mistresses and threw her off the balcony, killing her. Not a very kind dude by any stretch, which probably made the plan a lot easier to execute.

Interestingly enough, though Prince Leo was not a nice man, he was portrayed as being more liberal and progressive than his deeply conservative father, with potential plans to help the common folk. A nice shout-out to Prince Leo's real-life counterpart, Prince Rudolf, who also died by suicide in his hunting lodge.