To what extent can Marlowe's Doctor Faustus be viewed as a critique of arbitrary power?
It is very easy to read Marlowe's Doctor Faustus as a critique of arbitrary power to a significant extent. It is, however, not so easy to suggest that Marlowe intended his Faustus to be read and understood this way. In Marlowe's historical period, two things strongly argue against this reading.
The first is that the Christian religion had a ubiquitous presence and influence in Renaissance life, something akin to the presence and influence education has today: it is an all-pervading fact of life; it is the hope of a better life; it is the rock upon which all aspirations rest. Like education, religion was very difficult to get away from, and it was even more difficult to thwart its controlling influence over all aspects of life. Since life was ingrained with the powerful influence of religion, it is rare at this period that religion--one of two all-pervasive powers--might be lashed out at as an arbitrary power even though the Church at Rome had been powerfully and successfully challenged by the Protestant Reformation: the influence of Christian religion was still ubiquitous.
Arbitrary power is power that is not determined or confined by any law or regulation; it is determined by individual whim or will or desire alone; thus it is not predictable nor is it consistent. Religion at this era was governed, regulated, predictable, and consistent, even if not perfectly so and even badly so. Because of the ubiquitous nature of religion, the milieu wasn't one that readily yielded treatises or diatribes against the arbitrary power of religion.
The second thing is that Queen Elizabeth I was monarch of England. She was during her own lifetime counted as one of England's greatest monarchs. She was successful at bringing her people greater peace and prosperity than had been before. She was beloved as having courage and steely determination to rival a king's. It is also suggested by some historical records that Marlowe was in Her Majesty's employ as a secret agent and perhaps a spy. [There is also a theory that Marlowe's apparent early death was a staged act of espionage ordered by Elizabeth I to protect him from harm.] In light of these facts and half-facts [and speculations], it is highly unlikely that Marlowe would intentionally speak out against arbitrary power with the Queen in mind or with the danger of application being made to the Queen.
Therefore a reading of Doctor Faustus as a critique of arbitrary power would only reasonably be developed from a modern viewpoint without attributing intent to Marlowe. In such a reading, Faustus would be a symbol and metaphor for arbitrary power and the metaphoric outcome of arbitrary power as demonstrated by Faustus's arbitrary, ungoverned and unregulated striving after unlimited power.
All things that move between the quiet poles
Shall be at my command: ...
... can they raise the wind, or rend the clouds;
But his dominion that exceeds in this,
Stretcheth as far as doth the mind of man;
A sound magician is a mighty god:
Here, Faustus, tire thy brains to gain a deity.
I think the tortured body and soul of Faustus is a metaphor for the proud and independent spirit of man in the Renaissance. At the Renaissance, man's aspiring mind had become foremost in the preoccupations of the Renaissance. And yet, men were still under the sway if an arbitrary oppressive power: "Thou, traitor, Faustus, I arrest thy soul for disobedience to my sovereign lord. Revolt or I'll in piecemal tear thy flesh." Faustus's rebellion was an act of defiance. In the play, Bruno, the subversive 16th century philosopher was charged with heresy, a more than mortal sin and Faustus most certainly thwarted Rome's intention to have him arrested and executed. He shall be straight condemned to heresy // And on a pile of faggots bunt to death." Bruno was to be tried, convict and burned at the stake a few years later. In England, in 1612, Alexander legatt was the last heretic who was burned at the stake. A conspicuous allusion to another major figure of the Renaissance is : "I'll burn my books" which may refer to Paracelsus. He had this gesture of defiance to challenge the books about medecine and chemistry and the errors contained in them. He was exiled and may have been murdered too. Finally, Marlowe himself, a government spy, found himself in an impasse, experienced an aporia and the ending is often considered to be both paradigmatic of human condition and proleptic, symbolic of his own future death. Marlowe was arrested and charged with atheism and blasphemy. Although he was released, he met his death in a tavern in Deptford some time later, in May (?) 1593. His roommate was also arrested, tortured and accused Marlowe. Thomas Kyd died a year later, in dire poverty.