A hamlet is a small community within a town. Qualifications for a hamlet vary by country but the best example of a hamlet would be a suburb in a city.
It goes Hamlet - Smallest Village Town City CMA Megalopolis - Biggest
Hamlet is contemplating whether to live or die, facing a difficult choice.
City
The population of a hamlet is generally less than the population of a village
Hamlet - 1964 I was released on: USA: 23 September 1964 (New York City, New York)
Hamlet is from Denmark. The play takes place at the castle of Elsinore and it is implied that this is the permanent residence of the court and the Danish Kings. If so, his home "town" is Elsinore Castle, which isn't a town at all. They talk about "the tragedians of the city" who have left the city to come to Elsinore. The play is vague about where this city is, just as it is about Hamlet's home town, since it doesn't matter to the way the play plays out.
Elsinore...a port city in Denmark.
He who has a “why” to live can bear almost any “how”. – Frederich Nietzsche
They may do so in any city, town, village, hamlet or even alone in the wilderness.
A small town well outside the area of any city.
Hamlet is contemplating suicide. He is deciding whether he should kill himself, "not to be", or to live, "to be". He doesn't kill himself because it would be a sin.