Any help would be appreciated!

My g-g-grandfather Lawrence Greener Thompson (obviously anglicised unless his father was an Enlishman living in Denmark?) was born in Kobenhavn and jumped shipped in Melbourne in 1888 and lived happily ever after. According to his death certificate his father's name was George and his mother was Christina Pieterson.

As I don't speak Danish and I don't have anything other than a death certificate to go by I'm not sure where to begin. So any helpful resources would be greatly received.

I have tried the Danish club in Melbourne but unfortunately they were unable to help.

Thanks
Melissa