You can become a member of the Evangelical Lutheran Church in Denmark if you have received Christian baptism in another church, here or abroad. You do not need to be a Danish citizen to become a member of the church, but you cannot be a member of both the Evangelical Lutheran Church and another religious community at the same time. To become a member of the church, you will have to speak to a pastor. S/he will introduce you to the Evangelical Lutheran Church and help you fill in the membership form. Find your parish church and your pastor here.
When you become a member of the established church, you automatically belong to the parish in which you live. The whole country is divided into parishes, and each parish has its own church. If you wish to belong to a church in another parish, you can "loosen the parish ties", i.e. change parish without changing your address. You can do this by contacting the pastor in the parish that you would like to join.