Why is the Emperor so determined to have Anakin become his apprentice?

Well given that Anakin is the object of a prophecy that predicts a time of balance in the force, the Emperor would probably rather that the Sith are in control during such a time, and in which case he would need Anakin on his side to achieve that.

Palpatine sensed that Anakin was very powerful with the Force . . . more powerful than any other Force user so far. And he wanted someone like that associated with the Sith.

I think that as well as Anakin being a very powerful Jedi, and being the chosen one, The Emperor saw that Anakin was devoted to the ones he loved. He would die for them or worse. He would even turn to the dark side if that meant saving Padme. It's a bit Ironic how he ended up killing her when he was trying to save her though.

I think the prophecy of the Chosen One was part of the reason but he saw alot of potential in Anakin because of his extraordinary Force abilities. Anakin always had an aura of darkness in him with the fears he had of losing the ones he loved.

The emperor sensed Anakin was powerful with the force. He also had suspicions of him being the Chosen One.
So the Dark Side would grow more powerful if they had a strong fallen Jedi on their side.

I think that the Emperor took advantage of Anakin when his guard was down, the Emperor knowing that Anakin would do anything to save Padma me gave him something that he could not refuse. The Emperor is a lying backstabber.