Why is Claudius considered a strategist?
Claudius is a villian in this play, but his is exceptionally developed. He is a strategist because he predicts problems before they arise and set plans in motion to prevent them. He is a clever "monster," who is able to devise plots and plans that conceal his intentions and to manipulate others into furthering them.
Claudius has a number of foreign and domestic problems to contend with. One of the first internal problems is to have the country accept him as king. This is handled by having the Council support his marriage to Gertrude.
The Danish kingdom is threatened from without by young Fortinbras, son of the old ruler of Norway, who was killed by Hamlet's father. Claudius sends ambassadors to young Fortinbras's uncle (the brother of that country's dead king and presumably the current king of Norway), asking him to restrain his nephew and make him abide by the heraldic rules of the conflict between old Fortinbras and old Hamlet.
The king has noticed that Hamlet has been depressed since his father's funeral two months ago, and advises him that it is against heaven, the dead, and nature itself to continue immoderate grieving. Claudius names Hamlet as his immediate heir to the throne of Denmark and urges him to remain in Denmark as the ''chiefest courtier'' (I.ii.117) rather than returning to school in Wittenberg. Keeping him close allows Claudius to "keep an eye on him" and make sure he isn't a threat to Claudius' own power.