What do you want for/of/from/with me?

What do you want for me?

The first does not relate to the other three. It belongs in a different context. "What do you want me to have?"

What do you want of me?

What do you want from me?

These next two are essentially synonymous, meaning "What do you want me to give you?" The answer could be "help", "money", etc. These questions sound as if they are spoken by someone who is superior to the person he is talking to; someone in a position to give favors. For instance, a duke might ask this question of a stranger who has just come in his gates and wishes to speak with him. Another situation in which this might be asked might be one in which an underdog querulously asks it of his lord, as if in protest to his demands.

"From" is more common. "Of" sounds slightly archaic.

What do you want with me?

This one has a connotation of "What are you going to do with me?" It would not be asked by the duke (the person in a superior position) but by the stranger. For instance, when the stranger comes within the walls of the duke's castle, and is seized, and taken before the duke, he might well ask "What do you want with me?" and probably not "from me" or "of me".

That said, it is not only extreme situations in which this is used. "What do you want with me?" nowadays would probably indicate exasperation at being pestered for something.

Each phrase seems to have some understood words left out or replaced by want that change the meaning. Here are the phrases paired with possible equivalents:

What do you want for me?
What hopes and dreams do you have for me?

What do you want from me?
What do you expect me to do?

What do you want with me?
What business do you have with me?

What do you want of me is a less common expression, but it would have similar meaning to what do you want from me, the most commonly heard of the four. This from me phrase is often spoken in exasperation, as if the speaker doesn't feel able to actually offer anything in a given situation. Here is such an example: