A constitution is a statement of the basic principles and laws of a nation, state, or group, such as the U.S. Constitution. Another very common meaning of constitution is the physical makeup of a person. So think of a constitution as how a body (yours, the government's) is set up. If you have a strong constitution, it means you don't get sick very often.
I think the focus of the story is on Beowulf as a hero rather than a king because it is his heroic deeds that lead to him being offered the throne. Also to be king one must possess some type of heroic qualities in order to rule a land and its people.
The evil King of the Danes is Heremod, who is very cruel and tyrannical. He murders the people who support him, and his subjects banish him. Beowulf is contrasted against Heremod in the story, which uses Heremod as a juxtaposition of evil, whereas Beowulf is good and heroic.