I would start with the landing record. Is there a way to narrow down the time range, for instance through address or telephone registers? (Problem is, Victoria is a pretty big place in and of itself... I would concentrate on Melbourne first.)
By 1880, the landings in Perth should be fairly complete, and the way I understand it, that was often the first contact point where immigration was recorded. If you can find a landing record, you will have a better idea of his age, and of his state of origin, perhaps even, if you are lucky, the town. As it is, looking for a Karl Wilhelm Schiller, born 1850-1854 in Germany/Austria/Switzerland (or even England, due to his mother's last name) is not very promising.