The complete line is: "With mirth in funeral and with dirge in marriage." Claudius is trying to explain and justify his marriage with Gertrude which occurred only a couple of months after the death of her husband and his brother. Claudius is saying that he and Gertrude share mixed feelings because they are still mourning the death of King Hamlet while delighting in their love for each other. He acknowledges that there should have been a longer period of general mourning when he says: "...that it us befitted / To bear our hearts in grief and our whole kingdom / To be contracted in one brow of woe." But he suggests that it was somehow of great practical importance to the welfare of the kingdom for him to marry Gertrude as quickly as possible. He says that everyone approved of this decision: "Nor have we herein barred / Your better wisdoms, which have freely gone / With this affair along." It seems as if he feels he has to say something about this precipitate marriage but keeps it in vague generalities. Then he quickly shifts to practical matters, changing the subject that seems to embarrass him, as well it should. He starts talking about the warlike actions of young Fortibras and says no more about marriage.
In Act I, Scene 2, Claudius addresses his Lords and explains the reason he chose to marry his brother's wife, Gertrude, after the king's sudden death. Claudius appears to have the country's best interests in mind by consolidating his marriage and preparing to defend the nation against Young Fortinbras. Claudius is aware that his marriage may be met with mixed feelings throughout his court and mentions that he has the difficult task of balancing sadness with happiness. Claudius says,
With an auspicious and a dropping eye, with mirth in funeral and with dirge in marriage, in equal scale weighing delight and dole—taken to wife (Act I, Scene 2, lines 10-14).
Shakespeare uses antithesis to contrast Claudius's opposite emotions because the new king is both happy to be married but "sad" that his brother is dead. Claudius is essentially saying that he feels pleasantly amused during the funeral because of his marriage but laments at his wedding because the king is dead. Although Claudius is not actually upset about his brother's death, he knows it is necessary to give the illusion of grief so people will not become suspicious of him.