Lifeindenmark.dk gives you all the necessary information about what to do in case a person is deceased (in English)
Usually one will leave all practical aspects in connection with deaths and funerals to an undertaker. You can find a local undertaker by searching for undertaker (in Danish bedemand) on the internet. However, nothing prevents you from arranging a funeral or interment yourself.
If the funeral is to take place in the National Evangelical Lutheran Church (folkekirken), the deceased must be a member of folkekirken. Otherwise you can have a non-religious funeral.
If you need Copenhagen Cemeteries' customer centre will readily advise you about it all.
Read more about the funeral service the Copenhagen Cemeteries provide and find contact information at kk.dk (in Danish)