In The Tempest, what is a character sketch of Prospero as a Father, Duke of Milan and a Magician?

Your question correctly identifies the somewhat ambiguous
position that Prospero fulfils in this excellent play. It is Prospero of course who
plays the most significant role in the play - he is on stage longer than any other
character and he controls the fate of all the others on the island. You might want to
think about how he uses his powers and what that reveals about his character. His name
clearly means "prosperous" and you also might want to think about how prosperous he is
in the kindness he received from Gonzalo and his ability to win back his
kingdom.

Prospero initially lost his dukedom because he
failed to recognise his brother's raging ambition and he neglected his first duty - the
governing of Milan. Prospero was too busy studying magic to focus on his proper duties.
However, this situation has left him suspicious of human appearances. It may look as if
Ferdinand is in love with Miranda, but Prospero is determined to test the strength of
this love:

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They are both in either's pow'rs: but this swift
business

I must uneasy make, lest too light
winning

Make the prize
light.

It is uncertain about
how caring and loving Prospero is in his role of father. For example, he protests his
caring role as father to Miranda when he says to her:

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I have done nothing but in care of
thee,

Of thee, my dear one; thee, my
daughter

Although it was his
behaviour and ignoring of his duties that caused Miranda's fate in the first place.
Likewise ambiguity exists in his relationship with other characters. On the one hand he
protests that he loves Ariel "Dearly", yet then he clearly shows himself able to act
cruelly towards Ariel, bullying him and being violent towards
him.

However, he also shows an ability to forgive,
especially in the case of Alonso, Antonio and Sebastian. However, his harshness towards
Caliban must rate as his greatest weakness. Although Caliban did try to rape Miranda, it
appears his treatment of Caliban is unnecessary severe, and his act of giving Caliban
arthritis smacks of the excessive. This relationship between Prospero and Caliban is
extremely problematic, as by using non-human metaphors to describe and insult Caliban,
Prospero paves the way for treating him as not a
human.

Therefore, how you view Prospero depends a lot on
how you choose to play him. Is he a powerful and magnanimous man or a cruel sorcerer,
rejoicing in his power over others? The choice is yours...