Shakespeare uses a couple techniques to show the conflict between Hamlet and Gertrude in this scene, irony being the one most used.
First, Hamlet speaks in an aside (meaning no one else can hear him) to indicate he's not interested in speaking to his family -- they are "less than kind."
Then, Gertrude comments on Hamlet's clothing, indicating he's mourning too much. She tells him directly to be kind to Claudius. She says people die all the time, and he replies "aye, it is common," an ironic reply. The death of a king is not "common" -- nor is murder.
Then, Hamlet discusses the meaning of the word "seem," implying that people could fake their grief. (He's implying, perhaps, that Gertrude faked her grief.) His grief, however, IS real.