It's pretty simple actually. If the child is born in a local hospital, the live birth is automatically registered and the mother receives a record of live birth. This can then be taken to the national registry to receive the official birth certificate.
If the child is born at a place that is not a hospital, for example at home, the mother must take the child to the hospital or health center to receive appropriate medical attention. At that point, the mother is given a record of live birth which can be presented to the national registry to obtain the official birth certificate.
It's pretty much the same same as how it works anywhere else.
As far as citizenship, any child born in Cape Verde is automatically a citizen of Cape Verde. Dual citizenship is solely a matter of the laws of the country of citizenship of the parents. Thus, you should know the laws of your own country of citizenship BEFORE deciding to have a baby in Cape Verde.