We haven't been able to make CC available on the Kindle section of the Amazon app store, the only place where Kindles such as the Kindle Fire HD can obtain apps. Our suspicion is that Amazon feels that CC (and reader apps) would make a large hole in their garden wall. They also could be concerned about the support issues that would arise because of the difficulty of integrating with the built-in kindle app. Given that Amazon apparently makes almost no money on kindle hardware but instead makes money by selling content, one can understand their reasoning.
If you are brave and sufficiently technical then you could root your HD and install google play. However, given that an oops can lead to a bricked HD, this is something to be approached with a great deal of caution. We are not recommending this option. You could permanently break your Kindle, and even if successful you will probably void your warranty. This thread at xda-developers
discusses the process.
If you haven't yet purchased the Kindle, you might consider whether the advantages of Amazon's ecosystem outweigh the disadvantages of being inside their walled garden.