How do I monitor clipboard content changes in C#?

You can do this with pinvoke to the Win32 API AddClipboardFormatListener

The listener is a window handle (Form.Handle), and the form will be notified of changes with a WM_CLIPBOARDUPDATE notification

It is a more robust replacement for the older SetClipboardViewer API.

You could use SetClipboardViewer provided by Win32 API (through P/Invoke).

Here is a page which contains code to set one up in C#: http://www.codeguru.com/csharp/.net/net_general/tipstricks/article.php/c7315/

I've written up a small utility class that uses the AddClipboardFormatListener function function with a Message-only window to do just this.

/// <summary>
/// Provides notifications when the contents of the clipboard is updated.
/// </summary>
public sealed class ClipboardNotification
{
    /// <summary>
    /// Occurs when the contents of the clipboard is updated.
    /// </summary>
    public static event EventHandler ClipboardUpdate;

    private static NotificationForm _form = new NotificationForm();

    /// <summary>
    /// Raises the <see cref="ClipboardUpdate"/> event.
    /// </summary>
    /// <param name="e">Event arguments for the event.</param>
    private static void OnClipboardUpdate(EventArgs e)
    {
        var handler = ClipboardUpdate;
        if (handler != null)
        {
            handler(null, e);
        }
    }

    /// <summary>
    /// Hidden form to recieve the WM_CLIPBOARDUPDATE message.
    /// </summary>
    private class NotificationForm : Form
    {
        public NotificationForm()
        {
            NativeMethods.SetParent(Handle, NativeMethods.HWND_MESSAGE);
            NativeMethods.AddClipboardFormatListener(Handle);
        }

        protected override void WndProc(ref Message m)
        {
            if (m.Msg == NativeMethods.WM_CLIPBOARDUPDATE)
            {
                OnClipboardUpdate(null);
            }
            base.WndProc(ref m);
        }
    }
}

internal static class NativeMethods
{
    // See http://msdn.microsoft.com/en-us/library/ms649021%28v=vs.85%29.aspx
    public const int WM_CLIPBOARDUPDATE = 0x031D;
    public static IntPtr HWND_MESSAGE = new IntPtr(-3);

    // See http://msdn.microsoft.com/en-us/library/ms632599%28VS.85%29.aspx#message_only
    [DllImport("user32.dll", SetLastError = true)]
    [return: MarshalAs(UnmanagedType.Bool)]
    public static extern bool AddClipboardFormatListener(IntPtr hwnd);

    // See http://msdn.microsoft.com/en-us/library/ms633541%28v=vs.85%29.aspx
    // See http://msdn.microsoft.com/en-us/library/ms649033%28VS.85%29.aspx
    [DllImport("user32.dll", SetLastError = true)]
    public static extern IntPtr SetParent(IntPtr hWndChild, IntPtr hWndNewParent);
}

This class assumes that the notifications are always needed for the duration of the applications lifetime, however it could be modified to provide the ability to unsubscribe via the RemoveClipboardFormatListener function if required.