You can use both parts per million (ppm) and milligrams per kilogram (mg/kg) to describe the concentration of one compound dispersed in another. Scientists often use these units of concentration to provide information about the concentration of especially dilute solutions. The term "x" part per million means that out of one million units of total solution (for example, one million grams) there are "x" units of the compound of interest (in the case of the example, this would be "x" grams). The term mg/kg has a similar meaning, but uses specific mass units.
Ensure that the parts per million value you are converting is a mass per mass value. This is the typical usage of parts per million. If you are making a conversion to mg/kg (which are both mass units), the ppm value you are converting is most likely a mass per mass value.
Write out the numerical value of your parts per million measurement. For example, if your measurement value was 328 parts per million, you would write out 328.
Write the units mg/kg after your numerical value. In the case of the example, you would write out 328 mg/kg. This is all that is required because parts per million on a mass per mass basis are identical to milligrams per kilogram. You can prove this to yourself by considering that a milligram is one one-thousandth of a gram, or 0.001 gram, and a kilogram is 1,000 grams. So the ratio of milligrams to kilograms is 0.001/1000 which is 0.000001, which is 1/1,000,000.
TL;DR (Too Long; Didn't Read)
Since a liter of water at room temperature weighs almost exactly one liter, a concentration expressed in milligrams per liter of a dilute solution in water can also be directly converted to parts per million with minimal error.