Is there any evidence in Hamlet that this was not a tragedy of revenge, but rather it was a tale of motiveless malignity? Instead of revenge, was Hamlet's goal just to make life bad for everyone?

“The motive-hunting of motiveless Malignancy” is a comment written by Samuel Coleridge Taylor about Iago’s deep-seated hatred for Othello searching for rationalization. A similar analytical assessment provided by Stephen Greenblatt (in his book "Will In The World") is Shakespeare’s “strategic opacity”. In Othello Shakespeare has deliberately removed the explicit motivation for Iago’s malignancy that is present in Shakespeare’s source material; Giovanni Cinthio’s, “Un Capitano Moro”. The same can be said in Hamlet. The source material, Saxo’s, “Amleth”, is clear for example on Amleth’s actions, delay and feigned madness. Shakespeare has removed these from the play.
So, to the question of motiveless malignancy in Hamlet, I think the exploration could start near the end with Hamlet’s actions toward Rosencrantz and Guildenstern. There is no reason for Hamlet to condemn R&G to death. Simply, removing the death warrant and carrying out the mission as advertised would have been sufficient to thwart Claudius’ plan. Instead Hamlet goes the extra length to usher R&G to their deaths. Next, even if Polonius’ death is excused as a mistake (for the moment), the remorseless treatment of his remains is certainly questionable. Then look closer at Hamlet’s quest for revenge. Hamlet has prime opportunity to make Claudius pay for his deed while Claudius is trying to pray following his quick exit after “The Mousetrap”. Hamlet wants more than revenge. He wants revenge on Claudius' soul.
If we even look to the second scene of the play. The scene is one of joyous ceremony. A new king, and a marriage signal that the ship of state is no longer rudderless and that political stability is again at hand. And yet there is Hamlet; dressed in black with an aim to spoil the festivities under the guise of remembrance of a departed father.
There is a moment in the play where Hamlet is standing over the body of Claudius in Act 3 Scene 4.
"I do repent; but heaven hath pleased it so,To punish me with this and this with me, That I must be their scourge and minister.” (Emphasis added)
Was Polonius’ death a mistake? Would Hamlet not have recognized Polonius’ voice? Had Hamlet just left Claudius praying? There is a darkness to Hamlet that goes deeper than revenge. Good question though and worthy of exploration.

Your first step in researching this might be reading the beginning of the play. Hamlet is initially set on his course when the ghost of his father appears and demands that Hamlet kill Claudius, because Claudius had killed Hamlet's father and married Hamlet's mother.
Hamlet and his friends are initially skeptical about the ghost, worried that it is demonic or evil. Thus Hamlet is extremely hesitant about committing murder and refuses to do so until he has firm and incontrovertible truth that Claudius is indeed guilty. The death of Polonius is an accident.
Hamlet is given firm evidence of Claudius' intentions first by reading the instructions Rosencrantz and Guildenstern are given to kill him and then by Claudius' reaction to the play. Gertrude's death is due to drinking poison Claudius intended for Hamlet. 
Given that the character of Hamlet is distinguished by reluctance to act and a need to be assured that any action he takes is justified, the play quite clearly assigns revenge as his motive.