There is no way that the server can possibly read from the client's clipboard. So if you mean what the user has in their clipboard (at the server): it can't be done. You might be able to do some things with javascript, but then ASP.NET becomes largely irrelevant as a detail.
Alternatively, put it this way: have you ever seen this done?
Normally, to get a web-site to do something with non-text data, you need to use a file/upload control (Browse...) to send the data to the server. You might want to investigate this option.
If you need more power at the client, consider Silverlight, Flash, ClickOnce, etc (but try to avoid OCX - that is a dead technology).