What type of hoops do I have to jump through to launch high power rockets?
The FAR section that you are looking for is available on the ecfr website:

PART 101—MOORED BALLOONS, KITES, AMATEUR ROCKETS AND UNMANNED FREE BALLOONS

More specifically, it is Subpart C— Amateur Rockets

The main regulations that apply are (see the above link for the full text of the regulations):


  §101.23   General operating limitations.  
  
  (a) You must operate an amateur rocket in such a manner that it:
  
  (1) Is launched on a suborbital trajectory;
  
  (2) When launched, must not cross into the territory of a foreign
  country unless an agreement is in place between the United States and
  the country of concern;
  
  (3) Is unmanned; and
  
  (4) Does not create a hazard to persons, property, or other aircraft.
  
  (b) The FAA may specify additional operating limitations necessary to
  ensure that air traffic is not adversely affected, and public safety
  is not jeopardized.
  
  §101.25   Operating limitations for Class 2-High Power Rockets and Class 3-Advanced High Power Rockets.  
  
  When operating Class 2-High Power Rockets or Class 3-Advanced High
  Power Rockets, you must comply with the General Operating Limitations
  of §101.23. In addition, you must not operate Class 2-High Power
  Rockets or Class 3-Advanced High Power Rockets—
  
  (a) At any altitude where clouds or obscuring phenomena of more than
  five-tenths coverage prevails;
  
  (b) At any altitude where the horizontal visibility is less than five
  miles;
  
  (c) Into any cloud;
  
  (d) Between sunset and sunrise without prior authorization from the
  FAA;
  
  (e) Within 9.26 kilometers (5 nautical miles) of any airport boundary
  without prior authorization from the FAA;
  
  (f) In controlled airspace without prior authorization from the FAA;
  
  (g) Unless you observe the greater of the following separation
  distances from any person or property that is not associated with the
  operations:
  
  (1) Not less than one-quarter the maximum expected altitude;
  
  (2) 457 meters (1,500 ft.);
  
  (h) Unless a person at least eighteen years old is present, is charged
  with ensuring the safety of the operation, and has final approval
  authority for initiating high-power rocket flight; and
  
  (i) Unless reasonable precautions are provided to report and control a
  fire caused by rocket activities.
  
  §101.27   ATC notification for all launches.  
  
  No person may operate an unmanned rocket other than a Class 1—Model
  Rocket unless that person gives the following information to the FAA
  ATC facility nearest to the place of intended operation no less than
  24 hours before and no more than three days before beginning the
  operation:
  
  ...
  
  §101.29   Information requirements..
  
  ...