Basically you start with raw apple juice, preferably from your own apples. Let it sit in a jug with a bit of fabric fastened on the top with string or a rubber band to keep dust bunnies, cooties and bugs out, but let yeasty beasties in. It should turn to hard cider and eventually to vinegar. I stirred mine a few times (my deirections were to stir it once a day, I sort of did) and also had the dregs of some organic, unpasteurized unfiltered acv, which I added. It took about 3 months, but it turned out very well.
I have some dehydrated mother available, if you would like to try and see if it helps. I have never tried using dehydrated mother, and I accidentally dehydrated this nickle sized bit I found on the top of one of my bottles.
I used apples from a friend's tree. I cores and quartered them and "juiced" them using my blender. I added enough water to make it work. I put the resulting mess in a bucket, which I covered with an old clean tshirt. the next day I strained the juice. first using a colander , then using a jelly bag to get most of the solid out. Then I put it in some glass gallon bottles and covered them with a bit of cloth.
I have some wine that I hope will make some nice wine vinegar! I found a mother in a bottle of storebought organic vinegar and added it to the wine. I covered it with gauze and a rubber band and put it in a dark cupboard where I will check it in a month or so.
Basically, it is juice, then alchohol, then vinegar, whether you use a mother or not. The mother speeds it up.
So, how clean do you want the apple juice to be as far as being strained?
Does it need to be almost clear?
Do I need a mother to get started or will it work out by just using the yeasty beasties in the air?
The apple juice doesn't need to be clear to make vinegar, but I would strain it through a jelly cloth, or piece of muslin anyway. You can do that before or after the vinegar is made. And you don't need a mother if your apple juice is fresh and unpastuerized. Cover it lightly - again, a piece of muslin, old pillowcase, whatever - and leave it alone. Well no, not exactly alone; give it a shake or a stir when you think about it, to bring in a bit of oxygen. But if you don't think about it it (vinegar) will happen anyway. (PS - there a several threads already on this topic; the search function is your friend )