cheap travel from Amsterdam to Paris

Traveling from Brussels to Amsterdam and back to Paris would be backtracking.

You'd do best to fly into Amsterdam and take the train to Brussels and the train to Paris. Or start the route in reverse in Paris.

Between these three cities, the cheapest way is by bus (Eurolines), particularly if you're traveling soon.

Between Amsterdam and Paris, Thalys trains bought far enough in advance will almost match bus fares, if you can work with non-refundable tickets, and will be much faster. Trains bought closer to travel cost a fortune, and can even be sold out on some runs.

Trains between Brussels and Amsterdam can be Thalys or locals; the locals require changes and take longer, but will be much cheaper.

Websites for further research:

Eurolines (buses)

Thalys (Express trains between the three cities)

Belgian Rail International Page (for local trains to Amsterdam)

(I agree that if you can restructure your trip in a line from Amsterdam to Brussels to Paris, or vice versa, it would be easier, but I realize that may not be possible for various reasons).

Isbay, I agree with David. If possible, you should begin this journey in Amsterdam. As you describe the journey, you will probably pay an extra 58Euro or more in train fares. For a fairly low cost way to travel, you can purchase heavily discounted train tickets on https://www.thalys.com/be/en/ Tickets are non-refundable. I see fares as low as 29Euro from Amsterdam to Bruxelles-Midi. These special fares are limited so you have to look 90 days out from the time of travel and book early to get them. Secondly, I see fares as low as 22Euro from Bruxelles-Midi to Paris-Nord. You won't find fares much cheaper.

Megabus does a couple of runs a day, Brussels to Amsterdam for 12 euro, and Amsterdam to Paris for 12 euro.
http://uk.megabus.com/megabuscom.aspx