Basketry 101/Paper Coil Yarn Baskets
Back in the mid 80's, I was living in Alaska and was intrigued by the baskets the Eskimos made.  The only basket class available at the time was coil baskets, so I signed up.

Eventually, I entered my baskets in the annual Fur Rendevous craft competition and I won Division Champion for my work.  I was really proud of that ribbon!  I still have it hanging in my studio.

I thought I'd share with you the technique I learned.  Just recently, I picked up the materials and recalled how to make them. 

I hope you use your creativity to come up with ways to embellish them and form them to your liking.  You can easily finish a small basket in an evening, once you get the feel for it.

I've found another tutorial put out by the Craft Yarn Council with instructions on making a coil basket out of clothesline.  You can refer to the diagram they provide when wrapping the yarn.  I try to describe it as best as I can.  Hopefully, my pictures will help you.  Don't get discouraged.  This is really easy, just hard to explain.
What you'll need:

*Yarn--pick a strong yarn that doesn't break easily.  You can also use recycled yarn from sweaters.

*Paper Coil--or commonly called "paper twist."  You can purchase it at Michaels or any craft store.

*Tapestry needle

1.  Unroll the end of your paper coil like a fan about 2 inches. 
2.  Cut the paper from the bottom-up in an arch.
4.  Roll the end back up into a coil.
5.  Cut a workable length of yarn, thread your tapestry needle.  Lay the yarn on top of the paper coil and start wrapping.

6.  Wrap the coil with yarn about 1 1/2 inches.
7.  Now you're going to start forming the bottom of your basket.  Put down the needle and roll the end of the coil into a circle, keeping the yarn on the bottom of the basket.  Bring the needle up on the outside of the coil, then take the tip of the needle over and down through the middle of the coil.  Bring your needle back up the outside of the coil.  Bring the yarn up under the wrapping coil so the yarn is now on the inside.  Wrap yarn around coil 3 times.  Bring needle back down through previous coil.  This is called the Figure 8 as referred to in this link.  Please refer to the diagram in the provided link for clarification of this technique.  Continue wrapping coil in the manner until your yarn reaches 6 inches.

You will now need to add another piece of yarn to continue. 
8.  Adding yarn
Cut another piece of yarn a workable length. Place the yarn over the coil.  Wrap around 3 times and bring the needle down into the previous coil (Figure 8 Stitch).  Continue wrapping with the Figure 8 stitch over the previous yarn.  When the first yarn gets too short, remove your needle and thread on the new yarn you just added. 

Wrap yarn 3 times around coil, then use the Figure 8 stitch. 

Continue wrapping the coil using the 3 wrap/Figure 8 stitch method and adding yarn as you need to until the bottom of your basket reaches a size you want.  I suggest about 3 inches for your first basket.
Once you've determined how big the bottom of your basket will be, you'll want to start pulling the coil up to make the sides of your basket.

Flip the bottom over and look at where you started the coil.  This will be the starting point of pulling up the sides in order to keep it even.

When you've reached your starting point, position the coil on top of the yarn covered portion. 

Continue keeping each round even.  This is also the point where you can free form your basket.  Use your imagination.

To finish:

You'll want to anticipate where you'll want to end your basket.

Measure the paper coil out to the point you want to end and add 1/4 inch.  Cut.

Unroll the end of the paper as you did when you started.  See the pictures above in #1, 2 & 3.  Cut in a fan. Roll it back up into a coil. 

Continue the wrapping technique and Figure 8 stitch.  When you start to reach the end of the paper, about 1/2 inch, bring yarn OVER the coil instead of under.

Finish by stitching the yarn in between coils. Cut yarn off inside of basket.

Form a top the same way if you like, only you'll want to bring up the sides sooner. 
3. Discard the cut paper.  This is what the end should look like.
This picture shows the yarn that has been added.  The tail of the new length of yarn is on the right.  The tail of the first yarn is on the left. 
I clip the tails of newly added yarn as I go and wrap over the previous yarn. 
This picture shows how your yarn should be on the inside of the coil as you wrap over it.
These pictures show how you bring your needle down through the previous coil.  It's the Figure 8 stitch.
This picture shows the cut end ready to finish.
These pictures show wrapping the yarn OVER the end of the coil.
This picture shows the finishing of stitching the yarn between the coils before you cut it off on the inside.