Note that if $G$ be a self-complementary graph with $n$ vertices, then the following gives a self-complementary graph $H$ on $n+4$ vertices:
Let $H$ be the graph obtained by adding 4 new vertices $\{a,b,c,d\}$ to $G$, with edges $(a,b),(b,c),(c,d), (a,x), (b, x)$ for any vertex $x$ of $G$.

As there are self-complementary graphs on $1$ vertex and on $4$ vertices, it follows that, for any $n=0,1\quad (mod 4)$ there is a self-complementary graph on $n$ vertices.