A dresser needs to be sturdy enough to support about 60 pounds of clothing as well as the weight of its drawers. With a cardboard dresser, functionality is just as challenging, since they have to be able to move slide in and out of the dresser with the weight of the clothing.
To build the frame of the dresser you must determine the length, height, and depth desired for the dresser. Use a tape measure to measure the length, height, and depth. I recommend making the base and sides two boards thick, creating more support to bear the weight. Attach the two pieces for each side with wood glue. Attach the base, sides and top together with the wood glue.
To make the shelve bases, cut four pieces of cardboard to be the same length as the base and top. Glue two pieces together, so that you have two shelves of double thick cardboard. This can also be done with sets of three as each shelf base for increased strength. Then place glue on the depth end of each shelf. Slide them into place, having measured and marked their exact placement on the dresser.
Measure and cut the cardboard pieces for the shelves. The base of the shelf should be two pieces thick for weight support. For the pieces that are to serve as the front of the drawers, use a box cutter to cut rounded holes that are one inch tall and three inches wide to serve as handles. Attach all pieces with wood glue.
Before placing the drawers, line the bottom of each support shelf in the dresser with duct tape along the bottom edges. Do the same thing to the drawers themselves, including along the back. This will allow the drawers to slide easier in the dresser without creating friction.