How do we measure the concentration of a solution in "parts per million"?

You need the concentration in $g \cdot {L}^{-} 1$, and convert this to $\text{ppm}$. How?
$\text{ppm}$ as you know is parts per million; literally $\text{milligrams per litre}$, i.e. ${10}^{- 3} g \cdot {L}^{-} 1$ (ppm because there are a million mg in $1 \cdot L$ of water). It is typically used for trace quantities, and the density of the solvent (usually water) is almost identical to that of the pure solvent.