I thought I'd just throw this in to the mix...it makes things more complicated, not less, but some of you might like to think about it anyway.

As Sari has explained, Scandinavian Kings were elected, although the dying king often had his say in the matter. The same applied in England in Saxon times, and comes into the story of Ethelred the Unready, Harold Godwinson and William of Normandy. So Shakespeare would have been very familiar with the concept.

Old Hamlet was King, but in the sense only that he was a strong War leader. He was also, in his own right, a landowner, with a castle, lands, and probably a few manors extra. These posessions belonged to him alone, and were not tied into the Kingship, because the kingship location would change when a king was elected.

Now consider Claudius, the baby brother. As a a younger son, he probably has nothing at all, and is totally dependent on big brother. He may well have designs upon brother's wife, but apart from that he has to watch Old Hamlet reserve all his lands for his studious, indecisive , unwarlike young son. How would he feel about that? Pretty peeved, and worried about the family lands falling prey to another strong-man, possibly.

The old king dies, and Gert, as widow, has by law forty days to choose her "dower", i.e. the manor/land in which to spend her widowhood, before handing over the rest of Old Hamlet's posessions to Young Hamlet. Is Claudius to stand by and watch this happen? Not on your nelly! He moves, and quickly. He gets elected king, probably on the grounds of being a decent strong- man, and as sort-of-regent for Hamlet. That means Gert cannot be regent, even if she wanted to be. He also marries Gert...which negates her widowhood, and as we all know from understandng inheritance in JA, takes control of all of the Hamlet family posessions. So... Gert no longer has a dower. Young Hamlet has nothing to inherit. And, by making Gert "co-jointress" in ownership with himself, after their deaths, it is G&C's children that will inherit castle and lands, not Hamlet.