Solution 1:

The main problem you're having is you're trying to perform blocking actions in the Event Dispatching Thread. This will prevent the UI from been updated as repaint requests are not reaching the repaint manager until AFTER you've finished.

To over come this, you're going to need to off load the blocking work (ie the back up process) to a separate thread.

For this I suggest you have a read through the Concurrency in Swing Trail which will provide you with some useful strategies to solve your particular problem. In particular, you'll probably benifit from using a SwingWorker

Take a close look at doInBackground and the process methods

UPDATED with Example

Okay, so this is a REALLY simple example. This basically walks you C:\ drive to 3 directories deep and dumps the content to the supplied JTextArea

public class BackgroundWorker extends SwingWorker<Object, File> {

    private JTextArea textArea;

    public BackgroundWorker(JTextArea textArea) {

        this.textArea = textArea;

    }

    @Override
    protected Object doInBackground() throws Exception {

        list(new File("C:\\"), 0);

        return null;

    }

    @Override
    protected void process(List<File> chunks) {

        for (File file : chunks) {

            textArea.append(file.getPath() + "\n");

        }

        textArea.setCaretPosition(textArea.getText().length() - 1);

    }

    protected void list(File path, int level) {

        if (level < 4) {

            System.out.println(level + " - Listing " + path);

            File[] files = path.listFiles(new FileFilter() {

                @Override
                public boolean accept(File pathname) {

                    return pathname.isFile();

                }
            });

            publish(path);
            for (File file : files) {

                System.out.println(file);
                publish(file);

            }

            files = path.listFiles(new FileFilter() {

                @Override
                public boolean accept(File pathname) {

                    return pathname.isDirectory() && !pathname.isHidden();

                }
            });

            for (File folder : files) {

                list(folder, level + 1);

            }

        }

    }

}

You would simply call new BackgroundWorker(textField).execute() and walk away :D

UPDATED with explicit example

public class BackgroundWorker extends SwingWorker<Object, String> {

    private JTextArea textArea;
    private File sourceDir;
    private File destDir;

    public BackgroundWorker(JTextArea textArea, File sourceDir, File destDir) {

        this.textArea = textArea;
        this.sourceDir = sourceDir;
        this.destDir = destDirl

    }

    @Override
    protected Object doInBackground() throws Exception {

        if (sourceDir.isDirectory()) {

            // if directory not exists, create it
            if (!destDir.exists()) {
                destDir.mkdir();
                publish("Folder " + sourceDir.getName() + " was created");
            }

            // list all the directory contents
            String files[] = sourceDir.list();

            for (String file : files) {
                // construct the src and dest file structure
                File srcFile = new File(sourceDir, file);
                File destFile = new File(destDir, file);
                // recursive copy
                copyFolder(srcFile, destFile);
            }

        } else {
            try {
                copyFile(sourceDir, destDir);
            } catch (Exception e) {
            }
        }

        return null;

    }

    public void copyFolder(File src, File dest) throws IOException {

        if (src.isDirectory()) {

            // if directory not exists, create it
            if (!dest.exists()) {

                publish("Folder " + src.getName() + " was created");
            }

            // list all the directory contents
            String files[] = src.list();

            for (String file : files) {
                // construct the src and dest file structure
                File srcFile = new File(src, file);
                File destFile = new File(dest, file);
                // recursive copy
                copyFolder(srcFile, destFile);
            }

        } else {
            try {
                copyFile(src, dest);
            } catch (Exception e) {
            }
        }
    }

    public void copyFile(File src, File dest) throws Exception {
        // if file, then copy it
        // Use bytes stream to support all file types
        InputStream in = new FileInputStream(src);
        OutputStream out = new FileOutputStream(dest);

        byte[] buffer = new byte[1024];

        int length;
        // copy the file content in bytes
        while ((length = in.read(buffer)) > 0) {
            out.write(buffer, 0, length);
        }

        in.close();
        out.close();
        publish("File copied " + src.getName());

    }

    @Override
    protected void process(List<String> chunks) {

        for (String msg : chunks) {

            textArea.append(msg + "\n");

        }

        textArea.setCaretPosition(textArea.getText().length() - 1);

    }
}

Now to run...

new BackgroundWorker(textArea, sourceDir, destDir).execute();