It is difficult to speak generally about surrogate/probate clerks/courts because their rules and procedures (and fees!) vary according to location. There is no uniform way to approach the state vital statistics office or any county or local court, except for that one universal rule we family researchers must follow: be prepared! Check the web site of the agency or court. Call if no such site exists. Ascertain if there are procedures to follow or fees to pay to either view the documents in person or to order them delivered via mail.
The requirements are different for getting a death certificate from the state and getting a probate file from a surrogate/probate court or clerk's office. That is because in one case you are dealing with a state agency and in the other, you are dealing with a local or county court.
Not all states regard vital statistic records as public records. Many states are claiming state privacy concerns and laws as restrictions--hence, the relative ease of getting the probate file over getting a state certificate. The county courts and clerks are not subject to the same regulations that a state vital statistics office must follow.
That said, some courts have their own hoops to jump through. Quite a few use high fees and long waiting times. I have heard of courts in NY charging $30 just to allow you to use the index to find your docket number and file. Don't forget, mostly law firms use this stuff, so the fees are just a cost of business for them.
I always check out the court's website or call to see what their rules etc are before I go. As I stated, preparation is key!
Many courts will copy the file and send via mail, for a fee, of course.
But my main point is that alternative sources for death certificates do exist. It is important to find these alternate sources because states--and counties and countries--are making it harder and harder for family historians to obtain the records needed to establish their lineage.