What process do I follow to land a civilian aircraft on a military base?
Landing at a US military base isn't all that hard (I've landed at Ft. Drum before), you just have to get prior permission (PPR) (submitted at least 30 days in advance and confirmed within 24 hours according to 32 CFR 855.8), and need to have a "good reason" (as determined by them).  

32 CFR 855, Table 1 (which is quite long) includes the purposes that are normally allowed.

You will be required to submit the following forms to the base commander (See the A/FD):


DD Form 2400 - Civil Aircraft Certificate of Insurance
DD Form 2401 - Civil Aircraft Landing Permit
DD Form 2402 - Civil Aircraft Hold Harmless Agreement


32 CFR 855 contains the regulations pertaining to US Air Force airfields (and the rules are similar for each branch).

855.1 - Policy includes:


  (1) Normally, landing permits will be issued only for civil aircraft
  operating in support of official Government business. Other types of
  use may be authorized if justified by exceptional circumstances.
  Access will be granted on an equitable basis.
  
  ...
  
  (3) Any aircraft operator with an inflight emergency may land at any
  Air Force airfield without prior authorization. An inflight emergency
  is defined as a situation that makes continued flight hazardous.


855.5 - Responsibilities and authorities. includes:


  (6) Will not authorize use of Air Force airfields:
  
  (i) In competition with civil airports by providing services or
  facilities that are already available in the private sector.
  
  Note: Use to conduct business with or for the US Government is not
  considered as competition with civil airports.
  
  (ii) Solely for the convenience of passengers or aircraft operator.
  
  (iii) Solely for transient aircraft servicing.
  
  (iv) By civil aircraft that do not meet US Department of
  Transportation operating and airworthiness standards.
  
  (v) That selectively promotes, benefits, or favors a specific
  commercial venture unless equitable consideration is available to all
  potential users in like circumstances.
  
  (vi) For unsolicited proposals in procuring Government business or
  contracts.
  
  (vii) Solely for customs-handling purposes.
  
  (viii) When the air traffic control tower and base operations are
  closed or when a runway is restricted from use by all aircraft.