To get the DataGrid content to clipboard we should use the Clipboard GetData method.
Below example shows how to copy All the cells of DataGrid "myDataGrid" to textbox.
It will copy the ColumnName of DatGrid as IncludeHeader is selected for copy mode.
Me.myDataGrid.SelectAllCells()
Me.myDataGrid.ClipboardCopyMode = DataGridClipboardCopyMode.IncludeHeader
ApplicationCommands.Copy.Execute(Nothing, Me.myDataGrid)
Dim tempString as String = Clipboard.GetData(DataFormats.CommaSeperatedValue)
Me.myTextBox.text = tempString