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.
Beowulf is a man who boasts, yet he also has wisdom and humility. On the one hand Beowulf is reminded that pride will bring destruction: until his portion of pride increases and flourishes within him; then the watcher sleeps, the souls guardian; that sleep is too sound, bound in its own cares (5.9).