How do you clean cheesecloth?

For real cheesecloth, I throw it in the washing machine, with my clothes. It doesn't get sterile, but it gets clean enough that I don't worry about it becoming rancid. Rinsing it alone wouldn't suffice if you've been using it to strain fatty foods, like yogurt or cheese.

For the cheap "cheesecloth" that sold in the cleaning sections of grocery stores, that would shred to threads if you tried to clean it. It's cheap and disposable, so just throw it out.

To sterilise and degrease it, you can put it in a bowl of hot water with baking soda, and then steam it in a pan for half an hour, or better still, use a pressure cooker. Microwaving it works too but can be a bit messy - I do that to my dish sponges and small kitchen towels

Before the washing machine I put mine through the dishwasher, weighted or clipped to keep it out of the way of the spray arms and to spread it out. This gets the worst of the cheese (or tofu or whatever) off before the washing machine finishes the job, and after that sort of cooking I always have to run the dishwasher, while a suitable load of washing might have to wait.

In the washing machine I use a pre-wash with biotex unscented biological detergent, then a half dose of a minimally-scented detergent and an extra rinse.

I often give it a rinse in boiling water before use (with the aid of a wooden spoon) though not for jam or other things I'm about to boil.