Shakespeare's intention to introduce Hamlet through Claudius' monologue is first to make the audience aware of the recent series of events that have happened in court. Besides, it is a valuable technique to portrait Claudius' corrupt character. He tries to persuade everyone to move on after his brother's confusing death. Moreover, he attempts to manipulate people so that they overlook the fact that he has taken the throne and married his brother's widow.
Hamlet's soliloquy, on the other hand, is designed to display his inner feelings and thoughts aloud, without any other character listening but the audience.
