The influences upon Eliot in the making of his essay 'Hamlet And His Problems'.How does he use those influences in the criticism of Shakespeare's play 'Hamlet'?

I'm not sure exactly what Elliot's motivations for writing the article were but this excerpt sheds a bit of light on it.Hope it helps a little. I'll source-link it at the bottom.

Eliot used the theoretical perspective of New Criticism in order analyze the text only without any historical background to the Shakespeare getting the way. The text was analyzed to see if every idea, imagery, and revealing facts came alive that worked towards a unifying theme. T.S. Eliot used New Criticism to recruit contemporary literature scholars into studying the text and put aside any knowledge regarding Shakespeare to show its faults on a textual perspective.

Eliot pointed out that the text of Hamlet was so close to the works of Thomas Kyd's Spanish Tragedy and Arden of Feversham (T.S Eliot). All Shakespeare had to do was take the concept of revenge and make it his own by "revising the text of Kyd" (T.S Eliot). The only unifying theme that was the constant between Kyd and Shakespeare was revenge. The imagery and idea of Kyd's work became Shakespeare's when he began writing Hamlet.