Claudius murders King Hamlet

How Claudius went about the murder and why he, rather than Hamlet, was chosen as king is never explained but there are things commonsense would dictate. We don't know where Hamlet was when the murder was done but Claudius probably ensured he was not in Denmark, for the murder would be a futile act if the dead king were then to be succeeded by Hamlet rather than Claudius. More than likely Hamlet was at Wittenburg and under close surveillance by Claudius's henchmen. Presumably, Claudius had a foolproof plan with everything worked out exactly and nothing left to chance. Once the plan was ready, he had only to wait for some key event to happen to provide the opportune moment to kill the king. Timing was an essential factor to ensure success. Note the mention of timing in "The Mousetrap":

Lucianus: Thoughts black, hands apt, drugs fit, and time agreeing.

Rather than trying to work out or guess what Claudius did I will imagine that I am Claudius and reveal my plan:

Objectives to be met

100% certainty that I will be crowned king regardless of any Danish elective laws of succession. Absolutely no possibility of Hamlet being crowned, even if I have to kill him.
100% certainty that I will get Gertrude for my wife. I don't care whether she is chaste or wanton. Nor do I care whether she likes me or hates me or what she thinks. I am determined to get her.

Preconditions and the sequence of events to ensure my plan succeeds

Establish a team of henchmen who will, without question, say what I tell them to say, or lie, and do what I order.
Await an appropriate moment to kill King Hamlet.
Hamlet is at Wittenburg. I wont kill him, unless necessary, as this might drive Gertrude mad and I won't risk that.
Fortinbras (Norway) is training his troops to fight the Polacks.
I have been waiting for something like this.

The time has come to strike

I kill King Hamlet.
Naturally, the people are suspicious about the King's sudden death, until my henchmen give out a lie, saying that a post mortem examination discovered two fang marks on his leg (Achilles tendon) indicating a serpent had bitten him.
Ghost: 'Tis given out that, sleeping in my orchard,
A serpent stung me; so the whole ear of Denmark
Is by a forged process of my death
Rankly abused:
Hamlet's election as king is likely, however, because he is not in Denmark, I am appointed Regent pending his return.
My henchmen go to Wittenburg ostensibly to tell Hamlet to return urgently but actually to keep him there and make sure he is not informed until after I am elected king and married to Gertrude.
Other henchmen go to Wittenburg to recall Horatio and to France to recall Laertes and ensure neither contacts Hamlet.
I spread a false rumour that Fortinbras is about to invade Denmark using weapons of mass destruction!
......... young Fortinbras,
Holding a weak supposal of our worth,
Or thinking by our late dear brother's death
Our state to be disjoint and out of frame,
Colleagued with the dream of his advantage,
He hath not fail'd to pester us with message,
Importing the surrender of those lands
Lost by his father.
I order that our weapons' factories work day and night to re-arm Denmark and that we import more armaments.

Marcellus: And why such daily cast of brazen cannon,
And foreign mart for implements of war;
Why such impress of shipwrights, whose sore task
Does not divide the Sunday from the week;
What might be toward, that this sweaty haste
Doth make the night joint-labourer with the day:
Who is't that can inform me?
Horatio: That can I; At least, the whisper goes so.........
Fortinbras .....As it doth well appear unto our state
............. to recover of us, by strong hand
And terms compulsatory, those foresaid lands
So by his father lost: and this, I take it,
Is the main motive of our preparations
I order ambassadors to go urgently to Norway to complain of Fortinbras's (fictional) invasion threat.
.............. we have here writ
To Norway, uncle of young Fortinbras,—
Who, impotent and bed-rid, scarcely hears
Of this his nephew's purpose,— to suppress
His further gait herein; ......... we here dispatch
You, good Cornelius, and you, Voltimand,
Farewell, and let your haste commend your duty.
My national security henchmen warn the public that Denmark is in dire peril and an attack is imminent and that Hamlet's failure to return leaves the country leaderless at this critical time. It is imperative that someone be elected king.
Henchmen complain that Hamlet is only a student without military skills or experience of combat.
Henchmen name me as the most suitable person to lead Denmark. They praise my bravery and claim that though I recently fought alongside King Hamlet it was I who lead them in the final victorious charge against the English:
As my great power thereof may give thee sense,
Since yet thy cicatrice looks raw and red
After the Danish sword.
I will initially refuse the crown but accept it under a feigned protest — (just like Richard III).
I will say that the people love Hamlet, and if I am elected in his stead it will destabilise Denmark.
He's loved of the distracted multitude,
Who like not in their judgment, but their eyes;

the great love the general gender bear him;
Who, dipping all his faults in their affection,
Would, like the spring that turneth wood to stone,
Convert his gyves to graces
Henchmen will say that for the unity of the country, and because I have no children, that I name Hamlet as my successor to re-establish the line. Actually, I like Hamlet and am happy if he succeed me, in fact, I want him to succeed me.
Henchmen appeal to Gertrude to marry me so as to help unify Denmark at this critical time, mollify the pro-Hamlet lobby and consolidate his succession. Even if Gertrude doesn't want to marry me, this will persuade her to comply.
Gertrude might want Hamlet to attend the wedding but I won't take a chance on him stopping it, so it will take place immediately even if she thinks it is o'er hasty.
I marry Gertrude.
I am crowned king.
Hamlet returns.
Shortly after, on cue, my ambassadors return to announce that the invasion threat is over. It's a giant lie, of course, but nobody is ever going to check it out. I look like a brilliant statesman. Everyone is happy, except Hamlet, who sounds like a spoilt whining schoolboy.
Say, Voltimand, what from our brother Norway?
Voltimand: ................ he sent out to suppress
His nephew's levies; which to him appear'd
To be a preparation 'gainst the Polack;
But, better look'd into, he truly found
It was against your highness.
Fortinbras ..... receives rebuke from Norway, and in fine
Makes vow before his uncle never more
To give the assay of arms against your majesty.
Claudius: It likes us well;
I publicly name Hamlet as my son and successor.
........ let the world take note,
You are the most immediate to our throne .......
Our chiefest courtier, cousin, and our son.
Hamlet's cause is now dead and buried.