If you have a black-screen on the vSphere console (no matter if using .NET client, Web-Client, Remote-Console, etc.) typically the following things help you out:
- Having the correct port opened between your client and the ESXi host where the virtual machine is running on (TCP: 903)
- Correct working DNS: Your client must be able to lookup the ESXi FQDN
For sure there are many more things… most of them are well documented nowadays… so check them out.
There might be another reason if you use Windows 10 or Windows 2012 R2 (Not sure how it is in the older versions) where the symptoms (a black screen) might be similiar, but the problem is not vSphere related (even though it seems to be).