Molarity is a measure of number of molecules per unit volume, and ppm (parts per million) is a measure of proportion by mass. If the solution is dilute enough that adding the solute does not appreciably change the solution's volume or mass, you can perform this conversion. As a rule of thumb, once you get above 1,000 ppm, then parts per million is not the right unit of measurement.
Molarity is measured in terms of moles per liter (mol/L). Multiply this by the molecular mass of the solute in grams per mol (g/mol). The unit of moles will cancel, and you will be left with a measure of grams per liter (g/L).
Multiply your grams per liter (g/L) from the step above by the simple metric system conversion factor of 1000 milligrams per gram (1000 mg/g). The grams cancel, and the result is the concentration of solute in mg/L.
There are one million milligrams in one kilogram. A liter of water weighs one kilogram. Therefore, 1 mg/L equals 1 ppm. Your answer in mg/L is your final answer in ppm.