Hamlet and Prufrock?

Yes, "The Love Song of J Alfred Prufrock" is Mrs. Clinch's all-time favorite poem, but what does it mean? And why does it mention Hamlet?

"No! I am not Prince Hamlet, nor was meant to be;
Am an attendant lord, one that will do
To swell a progress, start a scene or two,
Advise the prince; no doubt, an easy tool,
Deferential, glad to be of use,
Politic, cautious, and meticulous;
Full of high sentence, but a bit obtuse;
At times, indeed, almost ridiculous—
Almost, at times, the Fool."

After reading it a second time and not being as lost as last year in Mrs. Clinch's class, I think I have an idea of what Eliot might be saying. It is obvious that Prufrock is a bit of a dork, with his balding head and his thin arms and legs. Eliot contrasts Prufrock with Hamlet to show his weakness and his insecurities. Hamlet is this great prince who defends himself and is popular. Prufrock is the complete opposite of that. He is self-conscious about the way he looks and he feels, "almost, at times, the Fool". He says he never could have been like Hamlet. He could have been a servant, to advise the prince, but he does not have the looks or charm or wit that Hamlet does.