In Shakespeare's Hamlet, did Hamlet's uncle really kill the king? How do you know? Is there evidence for this?

The short answer is yes, Hamlet's uncle really did murder Hamlet's father. The evidence for this is in Act 1, Scene 5.

In this scene, Hamlet's father appears as a ghost to his son. When he speaks to Hamlet, the king pitifully admits that he's doomed, for a time, to walk the earth at night and to suffer the fires of purgatory in the day until his "foul crimes" are "burnt and purged away." He then begs Hamlet to avenge his death and proceeds to reveal that he didn't really die from a snake bite, as claimed. In fact, he says that "the whole ear of Denmark / Is by a forgèd process of my death / Rankly abused." Basically, the whole country's been deceived into believing a lie, and the king thinks that the lie is a horrible trick to play on an unsuspecting populace. The words that proclaim the guilt of Hamlet's uncle are below:

Ghost: ...But know, thou noble youth,

The serpent that did sting thy father’s life

Now wears his crown.

Hamlet: O my prophetic soul! My uncle?

Ghost: Ay, that incestuous, that adulterate beast,

With witchcraft of his wit, with traitorous gifts—

O wicked wit and gifts, that have the power

So to seduce!—won to his shameful lust

The will of my most seeming-virtuous queen.

The king relates how he really died: his brother had poured henbane poison into his ear while he (the king) was sleeping in his orchard. The king tells Hamlet that the henbane caused his whole body to be crusted over with a "vile and loathsome" rash, and he reports that he died without receiving absolution for his sins ( "Unhouseled, disappointed, unaneled. / No reckoning made, but sent to my account / With all my imperfections on my head").

Before he leaves, the king begs Hamlet to make sure that "the royal bed of Denmark" doesn't become a "couch for luxury and damnèd incest." As for the queen, the king asks Hamlet to leave her alone; he tells Hamlet that his only aim should be to deal with his uncle. So, there's evidence (from the ghost's testimony) to substantiate the guilt of Hamlet's uncle, but there's no actual murder scene to show that Hamlet's uncle murdered the king.

Hope this helps!