What are the allowable ways to wire a kitchen range hood? (USA)
TL;DR: Put it on its own circuit or on a lighting circuit, but never on a kitchen receptacle circuit

Your options here are to put it on its own circuit (always works, but takes up panel space) or to put it on a lighting circuit (always works for hardwired hoods, may work for cord-and-plug connected hoods, ask your inspector for details).

Putting it on a kitchen receptacle circuit won't work

A receptacle within a cabinet and/or presumably greater than 5.5' off the ground falls outside the ambit of NEC 210.52 altogether, due to points 3 and 4 in NEC 210.52 proper:


  210.52 Dwelling Unit Receptacle Outlets. This section
  provides requirements for 125-volt, 15- and 20-ampere receptacle outlets. The receptacles required by this section shall be in
  addition to any receptacle that is:
  
  (1) Part of a luminaire or appliance, or
  
  (2) Controlled by a wall switch in accordance with
  210.70(A)(1), Exception No. 1, or
  
  (3) Located within cabinets or cupboards, or
  
  (4) Located more than 1.7 m (5 1∕2 ft) above the floor


As a result of this, it would violate NEC 210.52(B)(2) to place the range hood receptacle on a kitchen receptacle circuit, as Exception 2 does not apply to most hoods (the hood would have to be an integral part of the range for the exception to be usable):


  (B) Small Appliances.
  
  (1) Receptacle Outlets Served. In the kitchen, pantry, breakfast room, dining room, or similar area of a dwelling unit, the
  two or more 20-ampere small-appliance branch circuits
  required by 210.11(C)(1) shall serve all wall and floor receptacle outlets covered by 210.52(A), all countertop outlets covered
  by 210.52(C), and receptacle outlets for refrigeration equipment.
  
  Exception No. 1: In addition to the required receptacles specified by
  210.52, switched receptacles supplied from a general-purpose branch
  circuit as defined in 210.70(A)(1), Exception No. 1, shall be permitted.
  
  Exception No. 2: In addition to the required receptacles specified by
  210.52, a receptacle outlet to serve a specific appliance shall be permitted to be supplied from an individual branch circuit rated 15 amperes or greater.
  
  (2) No Other Outlets. The two or more small-appliance
  branch circuits specified in 210.52(B)(1) shall have no other
  outlets.
  
  Exception No. 1: A receptacle installed solely for the electrical supply to
  and support of an electric clock in any of the rooms specified in
  210.52(B)(1).
  
  Exception No. 2: Receptacles installed to provide power for supplemental equipment and lighting on gas-fired ranges, ovens, or counter-mounted cooking units.


Plugging it into a receptacle on a lighting circuit may or may not work, your inspector has to make a call on that

This option falls outside of NEC 422.16(B)(4) point 5:


  (4) Range Hoods. Range hoods shall be permitted to be cord-and-plug-connected with a flexible cord identified as suitable
  for use on range hoods in the installation instructions of the
  appliance manufacturer, where all of the following conditions
  are met:
  
  (1) The flexible cord is terminated with a grounding-type
  attachment plug.
  
  Exception: A listed range hood distinctly marked to identify it as protected by a system of double insulation shall not be required to be terminated with a grounding-type attachment plug.
  
  (2) The length of the cord is not less than 450 mm (18 in.)
  and not over 1.2 m (4 ft).
  
  (3) Receptacles are located to protect against physical
  damage to the flexible cord.
  
  (4) The receptacle is accessible.
  
  (5) The receptacle is supplied by an individual branch
  circuit.


However, since NEC 422.16(B)(4) says "shall be permitted", it basically means that the inspector must allow a cord-and-plug connected range hood that meets the parameters given, but may allow one that falls outside it; given that most reasonably-sized domestic range hoods draw less than 7.5A, having them share a branch circuit with other devices won't run afoul of 210.23(A):


  (A) 15- and 20-Ampere Branch Circuits. A 15- or 20-ampere
  branch circuit shall be permitted to supply lighting units or
  other utilization equipment, or a combination of both, and
  shall comply with 210.23(A)(1) and (A)(2).
  
  Exception: The small-appliance branch circuits, laundry branch
  circuits, and bathroom branch circuits required in a dwelling unit(s) by
  210.11(C)(1), (C)(2), and (C)(3) shall supply only the receptacle
  outlets specified in that section.
  
  (1) Cord-and-Plug-Connected Equipment Not Fastened in
  Place. The rating of any one cord-and-plug-connected utilization equipment not fastened in place shall not exceed
  80 percent of the branch-circuit ampere rating.
  
  (2) Utilization Equipment Fastened in Place. The total rating
  of utilization equipment fastened in place, other than luminaires, shall not exceed 50 percent of the branch-circuit ampere
  rating where lighting units, cord-and-plug-connected utilization
  equipment not fastened in place, or both, are also supplied.


This means that hardwiring them to a lighting circuit is also a viable option, provided your hood supports a hardwired connection. If I were to  cord-and-plug connect a hood to a lighting circuit, by the way, I would use a single receptacle instead of a duplex receptacle -- this makes clear that this is a dedicated receptacle for a single appliance, not a general purpose receptacle to plug whatever into.

Other hood tips

One of the biggest mistakes made with range hoods is oversizing -- even a "commercial" range in a residence doesn't need a commercial-sized hood capable of sucking well in excess of 500cfm of air out of the house.  Keep in mind that a commercial range hood is being used 6-12 hours a day, with the cooking appliance(s) under it being used continuously during that time, and has to meet OSHA standards for exposure to cooking smoke during that entire time because a line cook in a restaurant is exposed to that day-in, day-out, while your house has none of these issues to contend with.

Furthermore, a range hood that size needs special consideration regarding make-up air to avoid hazardous backdrafting of atmospheric combustion appliances in even a modestly tight house.  As a result, it's much better to use a smaller hood fan (200-300cfm is fine) with a hood system that has good capture efficiency and low static pressure loss in the ductwork than a larger hood with poor capture efficiency.