Our professor told us to do it by induction, for the cases $n = 0 \pmod 4$ and $n = 1 \pmod 4$.
We know, that for the case $n=4$, the only self-complementary graph is $P_4$ which is a path of four vertices. For $n=5$, it's either $C_5$ (a cycle of 5 vertices) or "the bull graph" (which looks kind of like a bull's head).
So now, by induction, we should find a way to construct a self-complementary graph for $n+4$ and $n+5$.
But I don't really see how we should be able to construct it. I also can't find anything on the internet about it.