As Liszt didn't leave a detailed program, the piece is certainly open to differing ideas as to its specific meaning. The works original purpose was as an overture to a dramatic production of the play, so there is no doubt musical references to events and people in the drama with a few references in the score as evidence of that. To paraphrase Liszt's thoughts on program music, he thoughts on it were explained by using the example of how a landscape could produce a mood within the viewer, and that music also could evoke a mood within the listener. As the landscape paints the mood, so can music paint the mood. So while some composers may have had a specific non-musical meaning behind their music, to me it is enough to know in general terms what the story is without a highly detailed, bar by bar analysis of which notes and phrases represent what specifically. As Liszt himself said in a letter to a friend:
Liszt did make some changes to the original overture and this final version was not heard for decades after his death. The work begins with the tempo indication Sehr langsam und Düster which loosely translates to Very slowly and gloomily. A horn makes the first entry with muted notes that sound unearthly. The orchestra enters after these notes and keeps with the eeriness of the music by playing softly, and the horn once again utters the stopped notes, orchestra responds as before for a few measures and then the time signature changes and the tempo indication changes to Moving, but moving very slowly. The indication Always gloomily appears occasionally throughout the first part. Tempo changes occur, Allegro appassionato, Allegro agitato, but the gloom never lifts off the orchestra completely. And as a reminder, Liszt repeats the opening tempo indications at the beginning of the ending, very slowly and gloomily with the added instruction Moderato-funebre , the death of Hamlet.
Someone once said that of all the major composers, Liszt was the one that threw his spear farthest into the future. That may be open to discussion, but Liszt did reveal the passions, terrors, loves and hatreds of humans in his music, perhaps to a degree as yet matched by any other composer. The symphonic poem is a mysterious and gloomy piece. Whether it follows the mood of the play, its specific actions, or if it 'paints' the moods and frames of mind of Hamlet, I leave to the listener.