= Solution
Let $q\in\mathbb C_\infty$ and let $G_q$ be its <stabilizer subgroup>. Choose a sufficiently small disc $U$ about $q$ such that
$$
gU\cap U=\varnothing
\qquad\text{whenever }g\notin G_q.
$$
Part (a) says that $G_q$ is cyclic and supplies a coordinate in which $U\to G_q\backslash U$ is $z\mapsto z^{|G_q|}$. Since the global orbit map identifies no additional points within $U$, this is also a chart on a neighbourhood of the orbit $Gq$ in
$$
R=G\backslash\mathbb C_\infty.
$$
Use these quotient charts at every orbit. On overlaps, a local inverse branch of one power map followed by a group translate and the other power map gives the transition function; it is holomorphic. The charts therefore define a conformal structure on $R$, and the local expressions $z\mapsto z^{|G_q|}$ show directly that the quotient map $\mathbb C_\infty\to R$ is holomorphic. This is the <finite conformal quotient of a Riemann surface> construction.
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