So. Took it home, disappeared in my work space for a while and came out with this beauty of a trash can!
Here's what I did:
Supplies: A simple plastic trash can with cut-outs. Fabric in two different colors. Scissors.
 Cut the fabric in 1-1.5 inch wide strips. It doesn't have to be very even.
 Fixate the first strip with a flat knot on the inside of the basket and start weaving!
 That's about it, actually. Weave until the whole space is filled and repeat for the other cut-outs with the second color. Hide the end of the last strip. Done!
Here's a simple trick how to combine two strips of fabric without getting a big ugly knot.
 Cut a hole into the end of your last strip and one end of your new one. With the new strip, pull the end with the hole through the hole of the other strip (like in the left picture).
 Then pull the long end of it through the second hole.
 Pull to tighten the knot. And there you have it: A nice and flat knot that will be almost invisible!