The Play Scene in Hamlet, Daniel Maclise (1842), Public domain, via Wikimedia Commons
(left) The Lion King, 1994
Often coloquially referred to as "Hamlet with Lions", this movie provides an interesting peek into some of the events preceding the play, including the contentious relationship between brothers Mufasa (Hamlet Sr.) and Scar (Claudius). In Hamlet, the brothers never share the stage, and therefore their relationship could range from as overtly jealous as the one in this clip to something else entirely, potentially involving Gertrude...
Hamlet V.2: Hamlet Kills His Uncle, the King Claudius, Gustave Moreau (data unknown)
Why did Claudius kill his brother? Why marry Gertrude? Why not kill Hamlet (Jr.) earlier?
The answers to these questions I defer to the actor.
That said, here's some potentially helpful context: lineage disputes in early modern England were common and often deadly. Securing one's political power was both messy and dangerous. Queen Elizabeth I's ascent to the throne followed the death of both her half siblings, and she was constantly undermined and her claim challenged by her political adversaries throughout her reign. Elizabeth's father, Henry VIII, had a string of wives whose fates ranged from divorce to death. Claudius's position would not have been easy to secure, and Hamlet (Jr.), in particular, could present a challenge to the throne if he wasn't so busy brooding...
Tales from Shakespeare: Hamlet, Charles and Mary Lamb, Public domain, via Wikimedia Commons
Hamlet Attempts to Kill the King, Eugène Delacroix (1834), CC0, via Wikimedia Commons
The play within a play (The Murder of Gonzago) in Hamlet is intended to catch Claudius in his guilt by forcing him to reveal his sin through his reaction. Hamlet (and Horatio) take Claudius's reaction to be a sign of his guilt, although this is not confirmed until he prays afterwards, a confession that Hamlet never sees. Suffice to say, the evidence of his guilt (to Hamlet) is a bit tenuous, textually.
A modern parallel of this sort of scrutiny could be found in some of the conspiracy theories surrounding the death of Princess Diana. Analysis of the royal family's compartment after her death have led some to speculate that her death was not an accident.
Let me not to the marriage of true minds
Admit impediments. Love is not love
Which alters when it alteration finds,
Or bends with the remover to remove:
O, no! it is an ever-fixed mark,
That looks on tempests and is never shaken;
It is the star to every wandering bark,
Whose worth's unknown, although his height be taken.
Love's not Time's fool, though rosy lips and cheeks
Within his bending sickle's compass come;
Love alters not with his brief hours and weeks,
But bears it out even to the edge of doom.
If this be error and upon me proved,
I never writ, nor no man ever loved.
Shakespeare's sonnets are rife with references to time's passage, the onset of old age, and the fading of beauty. They also often focus on love as a constant and unabatable force, such as Sonnet 116 (left). Hamlet's obsession over the speed at which Gertrude and Claudius married post-Hamlet Sr.'s death could be linked to these ideals regarding what love means.