We don't know much about Claudius's background, so you will have to work backward from clues in the play. We know that Claudius was the younger brother of the late King Hamlet, and we know from the play, at the end of act 3, scene 3 (where Claudius is praying), that he can't repent and ask forgiveness for the sins of murdering his brother and marrying his brother's wife. He can't repent, because he isn't sorry he did either one: he loves being king, and he loves being married to Gertrude.
A biographical sketch aims to paint a picture and to have a point of view, but it must be fact-based. It is typically organized chronologically, from birth to death. The focus, as you have indicated, is along the lines of the deceit. You might call it something like "The Deceitful Prince" or "The Deceitful Younger Brother."
It would be important to note that Claudius was a few years younger than King Hamlet. He probably grew up in his shadow, as Hamlet was destined to be king, and Claudius was the "spare heir." You could use evidence from what Prince Hamlet says (noting that it might be unreliable) to say that the late king was the one who shone like the sun: Prince Hamlet likens him to "Hyperion." We can understand how Claudius might be very jealous of his older brother, who not only got to be king but was handsome and athletic, got the beautiful girl, won the fight with Norway, and seemed to have it all. You could mention that the only way Claudius saw to get ahead was to become deceitful—he couldn't win by challenging his popular brother openly.
Since among the only facts we have is the story of how Claudius poisoned King Hamlet, you could talk about that and surmise that he bought the poison, waited until young Hamlet was away at the university, and then looked for his opportunity. You could then describe how the murder took place, as described by the ghost in act 1.
You can provide more detail about what happened after the murder, as then young Hamlet comes home and the action of the play begins. You would want to focus on how Claudius continues to use deceit, pretending to like and welcome young Hamlet, while simultaneously spying on him and being fearful of him. You would want to mention the deceit, after the Mousetrap play, by which Claudius tries to kill Price Hamlet, both through the letter he sends with him to England and by persuading Laertes to poison the tip of his rapier. You would definitely want to end with Claudius's plans backfiring and his being being murdered by the dying young Hamlet.
Good luck with this.