Claudius attempts to establish his popularity by justifying his actions and thus convincing the people that he is the best man for the position of King. First, he states how grieving for King Hamlet is good and right, but now they must move on and embrace his wedding to Queen Gertrude. He also speaks to Hamlet about his own relationship with his "son" and convinces Hamlet to stay at home where he and Gertrude can be near him. This, interestingly enough, is a very paternal speech to Hamlet and, therefore, establishes Claudius as a father-figure as well as compassionate and supportive.
What the reader should find ironic is that Claudius' behavior, marrying the Queen within months of the King's death and his appeal to Hamlet, are all self-motivated and extremely sneaky. Can he be trusted? Absolutely not.