Firstly

Don't use null layouts, they aren't helping you. Take the time to learn and use the available layout managers, see Laying Out Components Within a Container for more details.

Secondly

JLabel is a bad choice for trying to make background image. There's a few reasons, but lets just say they aren't really designed for the job

Thirdly

Start with a custom component and paint the background you want via it. Use it as the primary container for the JTextArea.

Configure the JScrollPane and JTextArea to be transparent and you will see the background through it

import java.awt.BorderLayout;
import java.awt.Dimension;
import java.awt.EventQueue;
import java.awt.Graphics;
import java.awt.Graphics2D;
import java.awt.image.BufferedImage;
import java.io.IOException;
import java.util.logging.Level;
import java.util.logging.Logger;
import javax.imageio.ImageIO;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JScrollPane;
import javax.swing.JTextArea;

public class Test {
    public static void main(String[] args) {
        new Test();
    }

    public Test() {
        EventQueue.invokeLater(new Runnable() {
            @Override
            public void run() {
                try {
                    JFrame frame = new JFrame();
                    frame.add(new TestPane());
                    frame.pack();
                    frame.setLocationRelativeTo(null);
                    frame.setVisible(true);
                } catch (IOException ex) {
                    Logger.getLogger(Test.class.getName()).log(Level.SEVERE, null, ex);
                }
            }
        });
    }

    public class TestPane extends JPanel {

        private BufferedImage background;

        public TestPane() throws IOException {
            background = ImageIO.read(getClass().getResource("/images/Paper.png"));
            setLayout(new BorderLayout());

            JTextArea ta = new JTextArea("This is some text");
            ta.setOpaque(false);

            JScrollPane sp = new JScrollPane(ta);
            sp.setOpaque(false);
            sp.getViewport().setOpaque(false);

            add(sp);
        }

        @Override
        public Dimension getPreferredSize() {
            return new Dimension(background.getWidth(), background.getHeight());
        }

        @Override
        protected void paintComponent(Graphics g) {
            super.paintComponent(g);
            Graphics2D g2d = (Graphics2D) g.create();
            g2d.drawImage(background, 0, 0, this);
            g2d.dispose();
        }

    }
}

As to the rest of you layout. Simple, make use of a number of containers/panels with their own layout managers and then bind the all together into a "master" panel with a suitable layout manager (BorderLayout looks like it would work)

After thoughts...

Depending on your needs, I might consider having some more logic in the paintComponent method to allow it paint more instances of the background if the container was resized beyond the size of the base background image, but that would depend on the image and your needs.

While more difficult, you could look at painting the background directly to the JTextArea (by overriding it), this would allow the background to scroll with the JTextArea, but would also require you to consider the point above