Well, there's no need to use bleach, which might actually be corrosive to the embedded brushes or the synthetic fibers of the snake. We're not trying to remove stains, but the build up of oils and bore crud.
Try either one of these two methods which work very well:
A) Put the snake in a mayonnaise jar with a couple of inches of mineral spirits (paint thinner). Shake vigorously for a few minutes. If it's really dirty, let it soak a while before you start shaking. Squeeze out the excess solvent and then blot it with an absorbent rag and hang up to dry. The dirty solvent can be left in the jar. The crud will settle out and the clearer liquid can be drawn off for the next use.
B) Same as above, but use hot water with a healthy dose of Simple Green or Krud Kutter or similar degreaser. Dispose of the dirty liquid and repeat with a hot rinse.