Fermentation to acetic acid
There are some recipes available on the web, my answer is based on this webpage and this information:
The process itself is rather easy and you can either use ready bought apple juice, homemade juice or apple cider (basically unfiltered apple juice). You will need some specially cultivated yeast, baker's yeast is generally not recommended since it might produce a lot of unwanted (and probably bad tasting) side products.
For the fermentation itself you need some (very) clean vessels in which the process can take place. These should be from a material which is not corrosive, as the vinegar would otherwise react with the vessel. Food-grade plastic, glass or stainless steel should be ok.
To have the reaction take place, the process need oxygen, otherwise the reaction will stop at the step of ethanol. This is also why acoholic fermentation is always sealed against the air, to avoid the oxidation of the alcohol the acetic acid. A cheese cloth will leave oxygen in, but keep insects and other contaminations out. If you need to filter the vinegar at the end of the process, a standard coffeefilter is good enough.
The following step by step directions come from this website:

Step 1 - Make or buy your apple cider or apple juice
Apple juice or apple cider is the starting point for making apple
cider vinegar. You can make your own, or buy it ready made, fresh,
bottled or frozen.  You can use a mechanical or steam juicer to
produce the juice / cider. To make your own juice (which is the same
as plain unfermented apple cider), see this page. Keep in mind that
you want to select sweet apples, like Fuji, Delicious, Mutsu, Gala,
etc. Green, unripe and unsweet apples (like Granny Smith) do not have
enough sugar to make good cider vinegar. Tips about selecting apples:
Apples used for cider don't have to be flawless. They do, however,
have to be free from spoilage. You can use blemished apples and small
sized apples. You can mix apple varieties together or use all one
variety. The only rule is to cut out any spoilage areas on otherwise
good apples. Spoiled areas will cause the juice to ferment too rapidly
and will ruin the cider. Don't use apples that appear brown, decayed
or moldy. Apples should be firm and ripe. Green, undermature apples
cause a flat flavor when juiced. The best cider comes from a blend of
sweet, tart and aromatic apple varieties. A bushel of apples yields
about 3 gallons of juice.
Step 2 - Make your starter yeast
Adding yeast to activate fermentation is not essential, but will speed
up the process and can produce a higher quality. Special cultivated
yeasts are available for this purpose at wine-making shops and
biological labs but bread yeasts are not recommended. To make a
starter, crumble one cake of yeast into one quart of cider and mix.
This makes enough starter for 5 gallons of cider; double the recipe
proportionately when making more.
Steps 3 - Making Alcohol and Acetic Acid
Pour all of the juice or cider into one or more containers to about
three-quarters capacity; do not close the lids on the containers.
Instead, cover the openings with cheesecloth secured with a rubber
band or string.
Step 4 - Store out of direct light, but some place with constant temperature
Stir the mixtures daily, making sure the cheesecloth is put back in
place. Keep the containers away from direct sunlight and maintain the
temperature at 60 to 80 degrees F.
Step 5 - Keep stirring daily, monitoring and tasting for the next 3 to 4 weeks
Full fermentation will take about 3 to 4 weeks. Near the end of this
period, you should notice a vinegar-like smell. Taste samples daily
until the desired strength is reached.
Step 4 - Filter Mother of vinegar
When the vinegar is fully fermented, filter the liquid through several
layers of fine cheesecloth or filter paper—a coffee filter works well
for this. This removes the mother of vinegar, shown at right,
preventing further fermentation or spoilage of the product. Mother of
vinegar is completely harmless and the surrounding vinegar does not
have to be discarded because of it. It can be filtered out using a
coffee filter, used to start a bottle of vinegar, or simply left in
and ignored.
Step 6 - Done! Storing Your Vinegar
The vinegar is now ready for storage in separate, capped containers.
Stored vinegar will stay in excellent condition almost indefinitely if
it is pasteurized. To pasteurize, heat the vinegar before pouring it
into sterilized bottles, or bottle, then place in a hot water bath. In
both cases, the temperature of the vinegar must reach at least 140
degrees F to pasteurize the product, and should not exceed 160 degrees
F. Use a cooking thermometer to ensure the correct temperature is met.
Cool the containers and store at room temperature out of direct
sunlight.