1-parameter NFW dark matter profile
You are missing one last ingredient: the correlation between the virial radius $r_{\mathrm{vir}}$ and the virial mass $M_{\mathrm{vir}}$. The idea is that the orbital time scale of a particle orbiting in the halo becomes roughly equal to the age of the universe at $r_{\mathrm{vir}}$. This implies a relation between the (mean) density of the halo and of the universe:
$$
    \bar{\rho}_{\mathrm{vir}} \equiv \frac{3M_{\mathrm{vir}}}{4\pi r_{\mathrm{vir}}^3} = \Delta \rho_{\mathrm{crit}}(z)
$$
where $\bar{\rho}_{\mathrm{vir}}$ is the halo's mean density interior of $r_{\mathrm{vir}}$, $\rho_{\mathrm{crit}}$ is the critical density of the universe (as function of redshift $z$), and $\Delta$ a parameter that depends on the cosmology. Often $\Delta=200$ is used and the corresponding quantities labeled $r_{200}$ and $M_{200}$ instead. See also this Wikipedia page.

So the way to construct a halo is as follows:


Pick $M_{\mathrm{vir}}$ (or $M_{200}$) and redshift $z$ (and your favorite cosmology)
Use above relation to find $r_{\mathrm{vir}}$ (or $r_{200}$)
Use the relation from Napolitano et al. (2005), or equivalent, to find $c$ and hence $r_s$
Obtain $\rho_{\mathrm{DM}}$ (if you still need it)