Burlap is cheap and easy to find, and simple to make into hanging basket liners as you can cut it with a pair of scissors. The best kind to use is the tightly woven type, usually used for sewing into sand bags or potato sacks.
The other kind that you might find in garden centers is meant to wrap shrubs in for the winter, and is a lot more loosely woven.
All the different kinds are made from jute, a natural fiber.
As all burlap is flimsy and easily rotted in contact with soil, it's crucial to also line it with poly film or plastic - garbage bags, or a shopping bag or dry cleaner bag - all work equally well.
To use burlap for a liner, cut it into a rough square, then put it into the hanging basket, then line it with the poly, cut into a circle. Stab the poly a couple of times close to the bottom for drainage.
The burlap is fine if it overlaps the top - you'll cut that off in a little while.
With it roughly in place lining the basket, start filling it (carefully) with your potting soil, gradually adding it in handfuls so as not to get it in between the layers of burlap and poly film.
Once you get to the top rim of the basket, adjust the burlap and then cut it off, leaving an inch or two to overhang.
If the chains to hang it aren't in place, hook those on now. Plant with your favorite flowers, hang it on a sturdy hook, and water well until the excess drains out.
Keep in mind that the use of burlap to line your hanging baskets is not permanent - they will rot quickly and must be discarded at the end of the season.
When you're done with them as a liner, put them on the compost pile to rot away.