To what extent is Hamlet a melodrama?

Let's look at the answer to the question this way: what are the essential characteristics of melodrama, and how does Shakespeare's Hamlet conform to those characteristics?
One feature of melodramas is sensational plots that revolve around unremitting tragedy, unrequited love, undeserved loss, or heightened emotion. Hamlet does involve these kinds of issues, but Shakespeare treats these issues on a high intellectual and emotional level.
Another feature is absurd, contrived plot lines and scenes. Unless you count the Ghost scenes, this is not a feature of Hamlet.
A hallmark of melodramas is single-minded, one-dimensional stock characters. The characters in Hamlet can be considered stock characters to some extent—Hamlet, the dashing hero; Ophelia, the beautiful, long-suffering ingénue and heroine; Claudius, the evil step-father and villain; Gertrude, the somewhat dim-witted, social-climbing mother; Polonius, the foolish, doddering old man; and Horatio, the hero's good friend and confidante. But the characters are intellectually and emotionally complex, nor are they one-dimensional or single-minded.
There is also an emphasis on good versus evil in melodramas. Good and evil aren't clearly defined in Hamlet. None of the characters, except possibly Claudius, are entirely good or entirely evil. As Hamlet says,

HAMLET: . . . there is nothing eithergood or bad but thinking makes it so. [2.2.257-258]

Intense emotion is present in melodramas. It is also present in Hamlet, but it is not exaggerated or manipulative. There are no contrived "tear-jerker" situations or scenes.
Melodramas often have an underlying social message or "moral of the story." Hamlet has no subplots or subtext regarding social or political injustice, the plight of the working man, struggling families living in poverty, or related issues.
Melodramas often have an episodic play structure. An episodic play structure involves a large number of different characters (like Hamlet) and locations (unlike Hamlet), covers a lengthy period of time (from 2-9 months, depending on who's counting, which is also true of Hamlet), and typically includes subplots in addition to the main story (also true of Hamlet).
Melodramas are written in a straightforward, uncomplicated, perfunctory style and are often written in stilted prose. Hamlet is written primarily in unrhymed iambic pentameter (blank verse), with some rhyming couplets to signal the end of scenes.
There is quite a lot of prose in Hamlet. The common folk, like the gravediggers, speak in prose. Hamlet speaks in prose to people below him on the aristocratic scale, and sometimes for dramatic effect, but the prose is hardly stilted.

HAMLET: . . . What a piece of work is a man! Hownoble in reason! How infinite in faculty! In form and movinghow express and admirable! In action how like an angel! Inapprehension how like a god! The beauty of the world, theparagon of animals! And yet to me what is this quintessenceof dust? [2.2.308-313]

Melodramas use spectacle, romantic scenic elements, and animals. Scene designers do what they can, but the "platform" and interior rooms of Elsinore castle, a plain somewhere in Denmark, and a churchyard aren't intrinsically very scenic or spectacular. There is the romantic element of the brook in which Ophelia supposedly drowned, but there are no lakes, forests, or other romantic locations. There are also no animals, unless the director decides to give Gertrude or Ophelia one of those yappy little dogs to carry around.
Sometimes melodramas have a musical background. This is not the case with Hamlet. Ophelia does some singing, as does the gravedigger, but their songs don't serve as background or underscoring to the play.
Both melodramas and Hamlet feature comic relief. In Hamlet, an example of this the gravedigger scene—but it's comedy on a high level, not exaggerated slapstick or buffoonery.
Finally, melodramas often feature fast action. In Hamlet, nothing really happens in a hurry except perhaps the fencing scene at the end of the play, and that's pretty much stop-and-go until everybody dies.
All in all, Hamlet demonstrates some characteristics of melodrama—which most plays do—but Hamlet is not a melodrama.