Q:

Did Vader/Anakin use the Force to kill Palpatine?

  Moments before he overthrew Emperor, Vader was badly beaten by his own son, Luke Skywalker. He lost his hand, and was described as being weaker than ever before. Yet, he somehow managed to rise, and after brief pause grabbed and carried Palpatine, for a few seconds enduring full strength of Force Lightning by arguably most powerful being in the Galaxy. Some people even compare him carrying Palpatine to Jesus carrying his cross. Both sacrificed themselves with great pain, to save the world. 
  Anyway, Anakin Skywalker was Choosen One of the Force, shatterpoint of Palpatine, and absolute shatterpoint of Sith. Yet question is, in his final act, did he use the Force, or better did Force help him to endure what othersfor control of the robed 
  figure despite the Dark Lord's weakened body and gravely weakened arm. 
  The Emperor struggles in his embrace, his bolt-shooting hands now 
  lifted high, away from Luke. Now the white lightning arcs back to 
  strike at Vader. He stumbles with his load as the sparks rain off his 
  helmet and flow down over his black cape. He holds his evil master high 
  over his head and walks to the edge of the abyss at the central core of 
  the throne room. With one final burst of his once awesome strength, 
  Darth Vader hurls the Emperor's body into the bottomless shaft.

The script does mention the use of the Force