Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.
Big Hint: The standard technique in all such differential geometry problems is to write $$ r = \lambda T + \mu N + \nu B$$ for some functions $\lambda$, $\mu$, and $\nu$. Differentiate, use what you're given, and use the Frenet equations and use the fact that $T,N,B$ form a basis to get three equations.