I dealt with this very briefly once upon a time here, when I discussed a couple of sources of the play. Thank you for giving me the opportunity to talk a little more about this topic. As you’ve been reading a bunch it’s likely you know that there’s no evidence Shakespeare read either Saxo Grammaticus’ tale or Belleforest’s French translation of the Amleth story, so it’s hard to speculate what the exposure might have been. Both versions were available in England when Shakespeare wrote the play, and with his level of compulsory education he would have been able to read in Latin or French. 

If Shakespeare did read either of these texts, it probably would have been Belleforest’s, as elements such as the prince’s melancholy in Hamlet are closer to that version. But, of course, that could have come from the so-called ur-Hamlet. Everything’s made difficult by the fact that we have no surviving copy of ur-Hamlet,a play that would have given some indication of what Shakespeare took from where. If ur-Hamlet was by Kyd there’s even more reason to think that a lot of the melancholy and troubled nature of Hamlet might come from there. Kyd's Spanish Tragedy (the quintessential Elizabethan revenge tragedy), features quite a lot of moral conundrums and a thoughtful revenger who’s troubled by the necessity for vigilante justice in the corrupt society he lives in

So I can’t offer any very useful ideas about what the influence of any Amleth would have been on Shakespeare’s play. It’s unsolvable, and some scholars would even say there’s no point in considering it. But this doesn’t mean it’s not worthwhile making a comparison between the Amleth character and Shakespeare’s Hamlet to see what Shakespeare’s focus was. 

Generally speaking, Amleth’s character is a lot simpler. He’s all about cunning and action. He’s a hero who has a noble goal and shows many of the qualities of a king (decisiveness, intelligence, outsmarting his enemies, even seducing his honeytrap to work for him). It’s also a success story. This doesn’t make the Amleth story less worthwhile. in fact, it’s hugely entertaining, and it gives the kind of satisfaction and closure that one might expect in a mythical tale about a hero. Amleth is an action hero of the Hollywood variety.

Shakespeare’s Hamlet is not at all an action hero. Shakespeare retains Hamlet’s capacity to do all the things Amleth does – Ophelia describes him as ‘a noble mind’ with 'the courtier’s, soldiers’, scholars’s eye tongue, sword…’ (3.1.149-0) – but makes the story about very different kind of hero in very different times. 

One of the major differences is the fact that Shakespeare's Hamlet is set in a Christian country, not the Nordic warrior society of the Historiae Danicae. Shakespeare was capable of writing largely secular works, as you can see from plays like King Lear, which is generally lacking in Christian references, and poems like 'Lucrece’ or 'Venus and Adonis’ which are written in the ancient Greek world of his characters. So the fact that he adds (or keeps ur-Hamlet’s addition of) the Christian dimension of the ghost returning from purgatory, and the reference to Wittenburg (suggestive of the reformation) suggests that he’s interested in depicting or exploring something different to the Amleth story.

To put it very simply, by making the murder secret and Denmark Christian, the focus shifts from action to contemplation, from public activity to individual consciousness. Shakespeare's Hamlet is more about what the hero thinks: his guilt, his doubt, and the troubled nature of revenge in a society that is Christian but also retains a lot of the ideas of a more heroic past (Hamlet’s father is always described as a heroic warrior, the kind who 'smote the sledded Polacks on the ice’ 1.1.62). It would seem that the tensions between different ideologies allowed Shakespeare to explore refracted identities, generation gaps, the ethics of revenge, and what counts as intelligence. 

This is a question about the sources of the play, and since Macbeth and Hamlet have considerably different sources, I’ll have to treat Hamlet quite differently.

You see, Macbeth is generally classed as a tragedy, but could just about count as a history, whereas Hamlet is definitely closer to tragedy. I’m afraid there’s almost no historical basis for the plot or the characters of Hamlet.

The main source for the Hamlet story is Saxo Grammaticus’s Gesta Danorium, which is a glorified history of Denmark. Saxo drew on actual history, Norwegian, Danish and Icelandic oral legends and tales, as well as rock and stone carvings. The story of Amleth appears to be one based on oral storytelling. When compared to other historical accounts from around his time, Saxo has a tendency to elaborate and exaggerate accounts. The equivalent of this kind of history in England might be the tales of King Arthur. In other words, it’s more legend than history. 

In Saxo’s version, the story of Amleth is about two brothers, Horvendill and Feng, who succeed their father Gervendill as the governors of Jutland. Horvendill slays the King of Norway in battle and is given Gerutha, the daughter of the (lengendary, 7th Century) King of Denmark, Rørik Slyngebond, in recognition of his battle fame. They have one child, Amleth. The Ophelia prototype is Amleth’s foster-sister. So there you have the main characters of the play. As you can see, they’re not the royal family, even if Hamlet is the grandson of the King of Denmark by his mother’s side. Also, Elsinore isn’t even in Jutland (or in Denmark in the 7th Century), so that’s a later addition.

Shakespeare doesn’t appear to have read Saxo’s version of the story. The tale was translated into French by François de Belleforest, and, since an English translation of the story wasn’t available until 1608, Shakespeare either read it in French, or based it on another version of the story. There appears to have been an earlier play, often called the Ur-Hamlet, possibly written by Kyd, that precedes Shakespeare’s Hamlet. There are no surviving copies of this play, but it’s quite likely that Shakespeare got his story from this play.

→ after graduating with an honors degree in physics, victor belleforest gets a job as an engineer at oasis springs’ landgraab nuclear power plant. he and his family move out from britechester to what is known as the desert paradise, only to find out that what he grew up seeing in travel brochures is actually 1% of what the city truly looks & feels like. now, almost completely broke from the across countries move-in, with his wife and two girls to take care of, plus a very sketchy neighbourhood to get used to, victor has never felt as stressed out and maybe even hopeless than now. all he wants to do is be able to get on the right foot again and change his family’s life.

→ as a stay-at-home mom, genevieve doesn’t have much perspective. people ask her why she, as clever and talented and young as she is, decided to live such an outdated life. genie never knows how to answer, but now there's a new life stirring within her womb and her distressed husband has no clue about it, so contemplating the question is the least of her worries.

→ evangeline is the perfect teenager: doesn’t swear, never stays up late, does her homework as soon as she arrives from school, helps her parents when they need her, plays with her younger sister when no one else has patience to do so, and the list goes on. but sometimes perfection could point to one engrained, major flaw, and evangeline’s might soon show itself in the shape of an addiction — to love.