Is correct that you need the solution density (weight of water) but usually, the people use to say ppm talking about mg/Liter (considering density =1g/mL). IsnĀ“t correct but you will find it in several times.
To convert the units, probably the links mentioned above will help you. You just must use the molecular weight of each compound and calculate how many mg do you have in a liter.
Sometimes, the nitrogenated species concentration use to be expressed in mg of nitrogen/Liter (i.e. mg N-NH3 / Liter means mg of amonnia nitrogen/liter). In this case, you must calculate how many mg of N are contained in 1 liter. (for instance 17g of NH3 contains 14g N, and so on with NO3 and NO2).
I hope it helps you.
Tripton