What do you do with the ground wire when connecting light kit to utility trailer?

Connect it to the trailer chassis. Anywhere is ok, just make sure it is connected to solid unpainted metal.

Please get a multi-meter and check resistance at the point where you connect ground wire from the towing vehicle and the points where you would connect the grounds for the lights. If the resistance are too high better use direct wire. I bought a folding trailer from local Harbor Freights and I had to use a ground wire because there were no conductivities in the frame.