Build you a frame of 2x4s or angle iron to fit inside the top lip of the bottom chest (or just on it if it doesn't have a lip) and the bottom cavity of the top chest. You may have to adapt for places where the wheels brackets were, but make sure the frame fits the top and bottom chests snugly. Temporarily fasten the frame to the bottom chest from underneath, set the top chest on, and drill holes in the top chest into or through the frame. Remove both and fasten the frame to the top chest.
It's all easy if you use a wood frame and wood screws. Drill and screw. If the frame is angle iron, spin nuts on medium size bolts and grind parallel flats in the end. Unscrew nuts to carefully reform threads. Put the bolts and washers down through the bottom of the angle iron frame (attached to the bottom of the top chest) and place so the the bolts go through the top of the bottom chest. Thread washers and nuts on exposed bolts and grab the flats with pliers to hold them while tightening.