We can gather from the information that we are given about Old Hamlet that he was a traditional king in that he was a warrior and a conqueror. We learn from Horatio that Old Hamlet brought Norway to his knees, forcing him to give a portion of his lands to Denmark. He also "smote the dreaded Poloacks on the ice." The spirit of Old Hamlet comes back wearing his armor, and the guards remark on his warlike prowess. Old Hamlet obviously settled disputes with forceful action. As a result, he was viewed as honorable and admired by his men.
Claudius is much more the politician. We see in his opening speech that he is glib and persuasive. He is able to put to rest the gossip about his hasty marriage to Gertrude by telling his audience that the marriage was a necessary political alliance that would make Denmark appear strong to its enemies: "the imperial jointresss to this warlike state." He hears petitions and solves problems diplomatically. When he learns of Fortinbras's planned attack, he dispatches ambassadors to the king of Norway to stop Fortinbras's attack. This method of solving problems through indirect means is Claudius' mode of operation. He typically enlists others to do his bidding as we see throughout the play and does not confront problems head on.
Of course, Hamlet sees major contrasts between the two in terms of their integrity and his relationship with them. He calls Claudius a "satyr," and his father "Hyperion." Later in Act 3, he tells Gertrude that his father had godly appearances, and that when she married Claudius it was as if she left a "fair mountain" and travelled to a "moor." But in terms of specifics, Hamlet is very as to how he and Claudius differed. We can assume that since Hamlet admired his father and from what we know of Claudius that the two differ greatly in moral character.
Hamlet ironically thinks more like Claudius. He is also persuasive, glib, charming and prudent. He thinks before he acts. He can play Claudius' game and outsmart his uncle at every turn. But Hamlet is essentially a moral figure, much like his father.