It depends precisely what you mean. An obituary is a piece of writing which sums up someone's life after their death, though usually it is written with an entirely positive spin. Though, obviously, when someone like Saddam Hussein dies, the obituaries don't take a positive spin. It depends.
I've put a link below for you which explains how to write an obituary generally. The basic, key point is that it's written in the third person ("He...") and that it goes in vaguely chronological order.
So you might want to make up something about Claudius' childhood (born in Denmark, surely?) and his early life, and decide whether he went to university. The play actually starts telling you about him when it tells you that he poisoned his brother in order to usurp his crown. After that, he married Gertrude at some point, and it's about then that the play's action begins.
What happens to Claudius during it? Well, he's preparing the country for some sort of war (cf Act 1, Scene 1). He has to cope with the problem of Hamlet - he brings in Rosencrantz and Guildenstern to spy on him. Hamlet kills Polonius, and Claudius packs Hamlet off to England. Claudius then persuades a vengeful Laertes to join with him in a plot against Hamlet (poisoned swords for fencing, and poison in a cup). After Laertes dies, Hamlet kills Claudius by forcing him to drink his own poison. Hamlet is king for all of one minute before he dies himself.
Hope that helps!