You can rinse a reusable coffee filter out in the sink.
After making coffee with the reusable coffee filter, simply dump the coffee grounds out into the trash and then run the reusable coffee filter under the sink in cold or warm water.
Getting some of the used coffee grounds down the sink drain is okay and won't clog your sink.
You can also place your reusable coffee filter in the dishwasher on the top rack to wash it.
Most times though a reusable coffee filter simply just needs rinsed and doesn't require any soap unless it has been left with old coffee grounds and grown mold.
Reusable coffee filters can save you money by not having to buy the paper filters all the time.
Reusable coffee filters can last several years depending on how much coffee you make and most often the reusable coffee filter will outlast the coffee maker itself.