This occurred while 'Determining packages to load'
access denied ("java.util.PropertyPermission" "deployment.user.cachedir" "read")
java.security.AccessControlException: access denied ("java.util.PropertyPermission" "deployment.user.cachedir" "read")
   at java.security.AccessControlContext.checkPermission(Unknown Source)
   at java.security.AccessController.checkPermission(Unknown Source)
   at java.lang.SecurityManager.checkPermission(Unknown Source)
   at java.lang.SecurityManager.checkPropertyAccess(Unknown Source)
   at java.lang.System.getProperty(Unknown Source)
   at org.lwjgl.util.applet.AppletLoader.getLWJGLCacheDir(AppletLoader.java:1064)
   at org.lwjgl.util.applet.AppletLoader$3.run(AppletLoader.java:1050)
   at org.lwjgl.util.applet.AppletLoader$3.run(AppletLoader.java:1038)
   at java.security.AccessController.doPrivileged(Native Method)
   at org.lwjgl.util.applet.AppletLoader.getCacheDirectory(AppletLoader.java:1038)
   at org.lwjgl.util.applet.AppletLoader.run(AppletLoader.java:843)
   at java.lang.Thread.run(Unknown Source)

<applet code="org.lwjgl.util.applet.AppletLoader"
        archive="lwjgl_util_applet.jar"
        codebase="."
        width="800" height="600">
 
  <param name="al_title" value="Targets">
  <param name="al_main" value="org.newdawn.slick.AppletGameContainer">
  <param name="game" value="javagame.Game">
 
  <param name="al_jars" value="Targets.jar, lwjgl.jar, slick.jar, lwjgl_util_applet.jar">
 
  <param name="al_windows" value="windows.jar">
  <param name="al_linux" value="linux.jar">
  <param name="al_mac" value="macosx.jar">
 
  <param name="separate_jvm" value="true">
</applet>

You included the wrong jars. LWJGL provides signed jars for applets. It's called "lwjgl_applet-<VERSION>.zip" and they are under the "basic" folder.

Thu Oct 11 22:07:12 BST 2012 ERROR:javagame.Game
java.lang.InstantiationException: javagame.Game
   at java.lang.Class.newInstance0(Unknown Source)
   at java.lang.Class.newInstance(Unknown Source)
   at org.newdawn.slick.AppletGameContainer.init(AppletGameContainer.java:124)
   at org.lwjgl.util.applet.AppletLoader.switchApplet(AppletLoader.java:1326)
   at org.lwjgl.util.applet.AppletLoader$2.run(AppletLoader.java:909)
   at java.awt.event.InvocationEvent.dispatch(Unknown Source)
   at java.awt.EventQueue.dispatchEventImpl(Unknown Source)
   at java.awt.EventQueue.access$200(Unknown Source)
   at java.awt.EventQueue$3.run(Unknown Source)
   at java.awt.EventQueue$3.run(Unknown Source)
   at java.security.AccessController.doPrivileged(Native Method)
   at java.security.ProtectionDomain$1.doIntersectionPrivilege(Unknown Source)
   at java.awt.EventQueue.dispatchEvent(Unknown Source)
   at java.awt.EventDispatchThread.pumpOneEventForFilters(Unknown Source)
   at java.awt.EventDispatchThread.pumpEventsForFilter(Unknown Source)
   at java.awt.EventDispatchThread.pumpEventsForHierarchy(Unknown Source)
   at java.awt.EventDispatchThread.pumpEvents(Unknown Source)
   at java.awt.EventDispatchThread.pumpEvents(Unknown Source)
   at java.awt.EventDispatchThread.run(Unknown Source)
This occurred while 'Initializing real applet'
Unable to create game container
java.lang.RuntimeException: Unable to create game container
   at org.newdawn.slick.AppletGameContainer.init(AppletGameContainer.java:148)
   at org.lwjgl.util.applet.AppletLoader.switchApplet(AppletLoader.java:1326)
   at org.lwjgl.util.applet.AppletLoader$2.run(AppletLoader.java:909)
   at java.awt.event.InvocationEvent.dispatch(Unknown Source)
   at java.awt.EventQueue.dispatchEventImpl(Unknown Source)
   at java.awt.EventQueue.access$200(Unknown Source)
   at java.awt.EventQueue$3.run(Unknown Source)
   at java.awt.EventQueue$3.run(Unknown Source)
   at java.security.AccessController.doPrivileged(Native Method)
   at java.security.ProtectionDomain$1.doIntersectionPrivilege(Unknown Source)
   at java.awt.EventQueue.dispatchEvent(Unknown Source)
   at java.awt.EventDispatchThread.pumpOneEventForFilters(Unknown Source)
   at java.awt.EventDispatchThread.pumpEventsForFilter(Unknown Source)
   at java.awt.EventDispatchThread.pumpEventsForHierarchy(Unknown Source)
   at java.awt.EventDispatchThread.pumpEvents(Unknown Source)
   at java.awt.EventDispatchThread.pumpEvents(Unknown Source)
   at java.awt.EventDispatchThread.run(Unknown Source)

package javagame;

import org.newdawn.slick.*;
import org.newdawn.slick.state.*;
/**
 * To-Do List:
 * 
 * Add life losses
 * Add disappearing target timing
 * Add more upgrades
 * Add more levels
 * Add different difficulties
 * Have better drawn graphics
 * Implement working sound
 */
public class Game extends StateBasedGame{
 
   public static final int menu = 0;
   public static final int level1 = 1;
   public static final int shop = 2;
   public static final int gameOver = 3;
   
   public static final String gamename = "Shooting";
   public static int ammo = 5;
   public static int maxlives = 3;
   public static boolean autoreload = false;
   public static String chosensight = "res/sights/sight1.png";
   public static int hiscore1=100;
   public static int hiscore2=75;
   public static int hiscore3=50;
   public static int hhiscore1=100;
   public static int hhiscore2=75;
   public static int hhiscore3=50;
   
   public static int level = 1;
   
   public static int sight2unlock = 0;
   public static int sight3unlock = 0;
   public static int sight4unlock = 0;
   public static int sight5unlock = 0;
   
 
   public Game (String gamename){
      super(gamename);
      this.addState(new Menu(menu));
      this.addState(new Level1(level1));
      this.addState(new Shop(shop));
      this.addState(new GameOver(gameOver));
      
   }
 
   public void initStatesList(GameContainer gc) throws SlickException {
      this.getState(menu).init(gc, this);
      this.getState(level1).init(gc, this);
      this.getState(shop).init(gc,  this);
      this.getState(gameOver).init(gc,  this);
      
      this.enterState(menu);
      
      
   }
 
   
   public static void main(String[] args) throws SlickException {
      try 
       { 
         AppGameContainer app = new AppGameContainer( new Game(gamename) );
         app.setDisplayMode( 800, 600, false );
         app.start(); 
       } 
       catch ( SlickException e ) 
       { 
         e.printStackTrace(); 
       } 
      
      
      
   }
}

Make sure your javagame.Game class doesn't use any parameters in its constructor method (i.e. it should be in the format "public Game()" ), otherwise Slick's AppletGameContainer can't create a new instance of it.

So get rid of "String gamename", you can just hard code it inside the Game contructor method if you want to set a gamename instead of passing one to it.

You can't use File to access resources from inside jar files (since they are not on the file system anymore but inside a zip file). So you need to use an InputStream to access them.