I finally figured it out. After I registered the host using "Register-CacheHost" I should have called "Add-CacheHost".
After I called "Add-CacheHost" I then called "Get-CacheClusterHealth". I finally got the results I expected.
The primary purpose was to create a Proof-Of-Concept in C# that would access AppFabric Cache on a local computer. When I executed the application which was running under my credentials I received the following error:
This is error seems to be a catch-all error. I've seen this error multiple times for different issues.
Since the application was executing with my credentials I needed to give my account permissions to access the cache. It did this by using "Grant-CacheAllowedClientAccount"
Now the cache is working with my application. I was able to get this to work on a computer that was attached to the domain and also to a computer that was in a workgroup.