Can I hang my bamboo fence on an existing chain link fence?

Yes. Make sure to check your local ordinances. Most cities and towns in the U.S. have a height limit on property line fences. The most common is 6' max, although decorative stuff that does not block views is generally acceptable above that. I have never heard of anyone having any problem with attaching a bamboo fence to a chain link or board fence, but your codes may be different.