What would Hamlet wear on stage?
On-stage, the actor playing Hamlet would likely be costumed like any other leading member of Shakespeare's theater troupe -- attired from a preexisting pool of garments purchased by the company, specifically from aristocratic garb more in line with the sartorial expectations on the upper classes than the presumed social class of the actor wearing them or the commoner standing in the audience. These garments would be purchased second- or third-hand by theater owners from garment dealers; gifts of clothing were common in wills, even from masters to servants, and garments inappropriate for a lower- or middle-class individual's day-to-day wear or even special occasion wear might be destined for a theater's costume closet in exchange for cash. This caused a little drama offstage -- upper-class responses to the abstract violation of social norms when you could see common actors swanning around dressed in emulation of their social betters and the concrete violation of sumptuary laws when actors left the theater premises in those costumes. The garments worn on stage undoubtedly experienced more wear and tear from use in performance than in their first life as street clothes, but they were still to Elizabethan eyes recognizable as too fine in quality for wear by individuals who were resolutely not real-life aristocrats. Writers like Stephen Gosson attacked this practice as one of the many perversities of Elizabethan theater -- if a low-born boy can stroll around snottily in silk and velvet as if he's rich, just because onstage he plays a prince, he's making a mockery of society's rules that separate the rich from the poor.
Overlashing in apparel is so common a fault, that the very hirelings of some of our Players, which stand at reversion of 6s. by the week, jet under Gentlemens noses in suits of silke, exercising themselves too prating on the stage, and common scoffing when they come abroad, where they look askance over the shoulder at every man, of whom the Sunday before they begged an almes.
This isn't quite the kind of cross-dressing we think about when we think about outrage over Elizabethan stage practices -- it's cross-class dressing, a potentially even more dangerous social transgression. Formal complaints were even lodged against one Yeoman of the Revels for permitting actors to leave theaters in costume. The very fact that older sumptuary laws were being referenced at all in this case suggests more than one person in a position of power was especially irked -- there weren't secret policemen lurking around every corner ready to launch themselves at you if they spotted a flash of gilt on your rapier hilt or some enamel aglets on your doublet points, but these sumptuary laws are an interesting, if by 1600 somewhat outdated, look at how clothing and attire were used as visual assessments of social status. The first actor playing Hamlet, Prince of Denmark was no dead-broke ruffian to begin with -- Richard Burbage was the son of a builder and had financial stakes in several theaters, in addition to profiting off his stage talents -- but he wasn't a prince either, he just got to dress like one.
Shakespeare uses the social signifiers of clothing and fine materials pretty frequently in the text of his plays, and we have sufficient textual cues -- in his history plays, in his comedies, in his tragedies -- that these signifiers would have been concretely represented on-stage. Despite the "period" setting of the play relative to Shakespeare's contemporaries, Hamlet would have worn what would then have been modern dress -- not with a particular visual convention to indicate a setting in antiquity.
Even allowing for the descriptions in the text of Hamlet's "customary suits of solemn black", it seems safe to assume that as a prince his clothing displayed the visual attributes of an upper-class Elizabethan man's clothing -- in terms of garments we're talking ruffs or falling collars, trunk hose or venetians, slashed and pierced doublets, likely carrying a sword or dagger; in terms of material we're talking silk, satin, damask, velvet, fur, potentially metallic lace (theatrical costumes seem to have gone wild for this) and so on. (From the text, we have an "inky cloak" in 1.2.) The cues to his social standing within the court at Elsinore would have been legible in his costuming to any audience member regardless of their own class -- you would have likely been able to visually assess at a glance the difference in rank between Hamlet and Osric, let alone Hamlet and one of the grave-digger clowns.
And we know from Ophelia's description that like nearly all Elizabethan men and women, he (customarily?) wears a hat:
[...] doublet all unbraced, / No hat upon his head, his stockings fouled, / Ungartered, and down-gyvèd to his ankle /
(Act 2, Scene 1)
Above all, the emphasis on Hamlet's black mourning garb suggests visual drama on stage. Another instance of off-stage showmanship in 1600 had the Earl of Essex changing out of all-black at his execution to a scarlet doublet in anticipation of his execution.
"the prysoner, who came in a gowne of wrought velvet, a blacke sattin sute, a felt hat blacke, a little ruffe about his necke, [...] he turned himselfe aside to put off his gowne [...] so opening and putting off his doublet, he was in a scarlet wastecoate [...]"
I certainly don't think Hamlet was consciously costumed like Essex -- that would have been politically prickly and chronologically implausible -- but there was a strong sense of visual drama in men's clothing at this time, and all-black men's clothing was not the norm. There was no minimalist impulse in Elizabethan men's fashion, even chromatically, so the visual impact of Hamlet's black garments on stage would have been strong, even if the actual wearing of them were limited to the first scene in which they're remarked upon.
What would Hamlet wear IRL?
What would Hamlet wear "in real life" in-universe, according to the sartorial standards of Elizabethan England? This is a little more speculative, but likely all of the above, minus the costume factor -- Hamlet is not "new money" or a lower-class male trying to ape his social betters. He's every inch a prince, "the glass of fashion and the mould of form" (3.1) despite his grief and diminished psychological state.
(One thing that always strikes me in discussions of how to costume Hamlet is his "sea-gown" mentioned in Act 5, Scene 2 -- maybe Elizabethan listeners would have reflexively understood Shakespeare's meaning better than I can. Is this a literal garment for wear at sea, is it sailor garb or a sort of wrapper, is it a poetic description of the mists and moisture rising from the ocean? When I encountered a modern director who brought up the latter interpretation it stopped me dead in my tracks, and to this day I'm still not sure what I'm meant to picture when I picture that sequence.)
What would Hamlet wear if he were actually a late Medieval Dane, say, circa 1450 or so?
That's a tougher question... and lucky for me it's explicitly not what you're asking about! I've never found any indication that either Shakespeare or the team that costumed Richard Burbage's Hamlet at the Globe were overly concerned with any kind of indication of earlier medieval "reality" on the stage, or even invoking some kind of national/ethnic difference between their fictional late-medieval Danes and the same characters if the play took place in a contemporary English setting.
And after that, as performances and depictions of the play continued through the 17th century and beyond, the physical interpretation of Hamlet's clothes varied with the norms of their era. Every stage production of Hamlet has to make their own decisions about what Hamlet's going to wear -- modern dress, an approximation of Elizabethan theatrical costume, medieval garb, medieval-inspired garb, something else? The "original practices" approach used by the Globe, approximating contemporary-to-Shakespeare stage aesthetics, or Ben Whishaw in a hoodie and a knit cap, or Maxine Peake in a white button-down, or this 1825 costuming guide for Hamlet, Hamlet Sr. and Gertrude, or Kenneth Branagh's vaguely 19th century Russian Hamlet, or Mel Gibson's more rugged quasi-medieval Hamlet, and so on. (This really is just barely scratching the surface of performance, I feel like 20th and 21st century Hamlets have worn pretty much every possible style of dress -- scruff mall Goth, 17th century Japanese aristocrat, school uniform, whatever.) But they're all working within their own imagination of the play's setting and the conventions of their own era of theater, even if the convention of their school of drama is to throw out all prior conventions -- whether that's "We won't have Hamlet in tights", or "we won't have Hamlet looking like a drippy teenager in a sweatshirt". But that's outside the scope of Elizabethan and Jacobean Hamlet-wear.
