What role did the RAT (Ram Air Turbine) play on US Airways flight 1549?
The full NTSB report can be found here and is the official source of information on the incident. According to that report Sullunberger did immediately start the APU which did deliver power and they determined they did not need to manually deploy the RAT. However the onboard systems deployed it automatically likely as a result of the initial power loss and sufficient airspeed (over 100Kts). 


  Regarding step e, the pilots stated that they determined that
  electrical power was established and, therefore, that the RAT did not
  need to be manually deployed. Further, immediately after the loss of
  engine thrust, the captain started the APU.


This can be heard in the official transcript 


  At 1527:14, the first officer stated, “uh oh,” followed by the captain
  stating, “we got one rol- both of ‘em rolling back.” At 1527:18, the
  cockpit area microphone (CAM) recorded the beginning of a rumbling
  sound. 
  
  At 1527:19, the captain stated, “[engine] ignition, start,”
  and, about 2 seconds later, “I’m starting the APU [auxiliary power
  unit].”5  
  
  At 1527:23, the captain took over control of the airplane,
  stating, “my aircraft.”


However the RAT was found to be deployed (due to the automatic systems on the aircraft) when the airframe was recovered 


  When all of the airplane’s electrical power is lost and the airplane
  has an airspeed greater than 100 knots (kts), the RAT will
  automatically deploy and begin providing electrical power. The RAT was
  found in the extended position when the airplane was recovered from
  the water.