With a filler form, set to use C#, that has a using System.Windows.Forms statement at the top of the code file, add code to your button:
string text = this.MainDataSource.CreateNavigator().SelectSingleNode("/my:myFields/my:field1", NamespaceManager).Value;
Clipboard.SetText(text);
See the part in bold that says "/my:myFields/myfield1"?
You need to change that to the correct XPath for your field. Find the field in the fields task pane in InfoPath designer. Right click and select Copy XPath, then paste it in.