(I also don't use plastic food dishes. Only ceramic or stainless steal).
It looks like acne to me! From what I've read, you shouldn't use soap to clean it, as it can leave residue that can further clog the pores and exacerbate the problem.
I personally use a product called hibiclens, that you can get at pharmacies. I've read a lot of positive stuff about it for feline acne, and I actually used it on my piercings myself. It works to help clear up acne on my two girls quickly, one of them gets it quite often. I use it by diluting it roughly 1 part hibiclens 20 parts water (so about 1/2 teaspoon in 3 Tablespoons of water). I have 3 cotton balls or gauze squares- one with the diuted hibiclens that has been squeezed so it doesn't drip, one with plain water, and one dry. I clean carefully with the hibiclens, follow up with the water one to "rinse", then dry well with the dry one.
You can use rubbing alcohol, but your cat may not like it due to the smell (kitty noses are very sensitive) and burning feeling. Or you can use hydrogen peroxide. If it doesn't improve with those methods, there is a topical medication you can get from your vet. I've never had to resort to that, though.
Keep your bowls clean as possible, especially until it clears up. I clean them once a day when fighting the acne. Some people say that the soap residue on bowls can add to the problem, and opt to clean their cat's bowls with scolding hot water, or with something like a lysol wipe (and rinse well with hot water after).
If your kitty gets wet food or has regular feeding time, clean their chin after feeding. If you do free feeding, clean it twice a day. I hope your kitty's chin clears up soon! :)
Avoid using any of the Lysol brand products. The main ingredient in many of the products is a chemical (phenol) that is very toxic to cats. Avoid other cleaning products that contain any sort of pehnols.
Hot and water and soap are fine to clean out the food and water bowls. You could even soak the bowls in a diluted bleach solution and then wash in hot water and soap.
Just keep your cat's chin clean. Wiping with a damp towel helps. You can also use baby wipes or pet grooming wipes. Some people like to swipe an acne cleaning pad (the kind Human teenagers use. Stridex is one brand) on the acne area to help clear it up. The black specks can be gently removed by running a fine toothed comb backwards through the fur.