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.