I was a detailer for a few years when I was younger and it’s been a hobby of mine ever since. First things first, try not to wash in direct sun, especially for dark colours, like black vinyl. For a soft vinyl tonneau cover I would use a diluted all purpose cleaner like Simple Green (dilute based on strength needed) with a soft bristle brush. Keep the paint of your truck wet and wash or rinse the cleaner really well from your paint. If it’s too strong it can leave streaks that will need to be waxed or polished out. After the cover is rinsed really well, let it dry. Then get yourself a good quality exterior vinyl dressing. I prefer PRO products if you can find them in your area. Spray it onto a wax applicator and then massage it into the vinyl in small circles. Depending on how faded and thirsty your vinyl is it may need a few coats. Once it’s looking good and even you can either leave it as it is for the high gloss look but that will most likely gather dust. Get a clean rag and give the vinyl a wipe down to get any excess dressing off. This will help prevent dust. A nice wipe down can get you a nice semigloss look that will last for weeks (depending on the product you use).
FYI
You can use this process for tires and all that exterior textured black plastic trim (great for getting dried wax off of trim). Just vary the amount of cleaner you use depending on what your cleaning. And always remember to wipe down your tires after applying any kind of tire shine no matter what the bottle says. Otherwise as soon as you drive away it’s gonna fling all over tour freshly cleaned paint.