If we're car camping, we have an ash bucket we bring. Just make sure the bottom is thick enough to handle coals.

Let it burn to ash and dump it in the firepit.

Don't leave fires unattended. Douse, stir, and feel before leaving.

If you are primitive camping, a big bucket of water and a spade or sturdy stick are your friends. Thoroughly douse the embers, break up the coals, douse it again for good measure, then scatter any remaining cool ashes outside the campsite.