CardLayout (swing) with action listeners on buttons?

Question

Problems:

Your code doesn't compile for us since we don't have the JLetterField class.
You're trying to add the JFrame's contentPane back on itself, and causes an exception and doesn't make sense.

Edit 1:

Your latest code shows you putting everything into one very large run() method but in doing so, you loose much and gain nothing.
I suggest getting rid of the Runnable interface, there's no need for it, and creating a true OOP compliant class, one with private fields and public and private methods.
Your actionPerformed method shows no attempt at changing the CardLayout's displayed card.
I suggest that you get rid of the code to show a new window and try to place card swapping code there.
Make your CardLayout and the card-displaying JPanel fields of the class so that other methods can access them and call their methods.

Edit 2:

For example the following code shows the swapping of cards using 3 JButtons. One to get the previous card, one to get the next card, and one to show how to get a specific card (here the 2nd):

import java.awt.BorderLayout;
import java.awt.CardLayout;
import java.awt.Dimension;
import java.awt.GridLayout;
import java.awt.event.ActionEvent;

import javax.swing.*;

@SuppressWarnings("serial")
public class CardLayoutEg extends JPanel {
   private static final String[] CARD_LABELS = { "one", "two", "three", "four",
         "five", "six", "seven", "eight", "nine", "ten" };
   private static final int PREF_W = 400;
   private static final int PREF_H = PREF_W;
   private CardLayout cardlayout = new CardLayout();
   private JPanel cardHolder = new JPanel(cardlayout);
   private Action[] actions = { new ShowPreviousAction(), new ShowNextAction(),
         new ShowTwoCardAction() };

   public CardLayoutEg() {
      for (String cardLabelText : CARD_LABELS) {
         JLabel cardLabel = new JLabel(cardLabelText, SwingConstants.CENTER);
         cardHolder.add(cardLabel, cardLabelText);
      }

      JPanel btnPanel = new JPanel(new GridLayout(1, 0, 5, 0));
      for (Action action : actions) {
         btnPanel.add(new JButton(action));
      }

      setLayout(new BorderLayout());
      add(cardHolder, BorderLayout.CENTER);
      add(btnPanel, BorderLayout.SOUTH);
   }

   @Override
   public Dimension getPreferredSize() {
      return new Dimension(PREF_W, PREF_H);
   }

   private class ShowPreviousAction extends AbstractAction {
      public ShowPreviousAction() {
         super("Previous");
      }

      @Override
      public void actionPerformed(ActionEvent e) {
         cardlayout.previous(cardHolder);
      }
   }

   private class ShowNextAction extends AbstractAction {
      public ShowNextAction() {
         super("Next");
      }

      @Override
      public void actionPerformed(ActionEvent e) {
         cardlayout.next(cardHolder);
      }
   }

   private class ShowTwoCardAction extends AbstractAction {
      public ShowTwoCardAction() {
         super("Show Two");
      }

      @Override
      public void actionPerformed(ActionEvent e) {
         cardlayout.show(cardHolder, CARD_LABELS[1]);
      }
   }

   private static void createAndShowGui() {
      CardLayoutEg mainPanel = new CardLayoutEg();

      JFrame frame = new JFrame("CardLayout Example");
      frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
      frame.getContentPane().add(mainPanel);
      frame.pack();
      frame.setLocationByPlatform(true);
      frame.setVisible(true);
   }

   public static void main(String[] args) {
      SwingUtilities.invokeLater(new Runnable() {
         public void run() {
            createAndShowGui();
         }
      });
   }
}