Try cracking the lid of the grill, yes you want smoke, but you also want airflow to allow the smoke to roll across the surface of the cheese. As tom b stated an ice pan directly under the cheese will help.
Maybe consider starting the smoking process in the early morning hours when it is cooler and the sun isn't heating up the grill.
When sorting thru all of the basics of cold smoking, the main fact is that you are cold smoking so the cheese will not sweat or melt from heat.
The term "cold smoking" usually refers to smoking at temperatures of 90 degrees or less, something that can be quite difficult with the backyard smoker on a sunny day. Some books say that cheese should be smoked under 80°, but you will probably want to keep your cheese below 70° so it will not sweat, and if you go above 90 degrees you could possibly end up with some of the softer melted cheese.
I Personally like to keep things about 50-55° when working with cheese. You set the cheese on racks above trays / pans of ice into the cooker to keep the temperatures down during smoking. You can also take advantage of cooler outside temperatures to help achieve the lower temperatures associated with cold smoking. I do a lot of cold smoking in the fall, winter, and early spring when conditions are near perfect for cold smoking.