No, all your characters will stay the same and ovcoarse you can keep leveling them. The battle chest just gives you the ability to keep playing more of the game and for longer. Same for any expansion. I used the same thing as you when I started wow, so have fun!
You need to purchase the monthly sub to level your characters past 20, purchasing the expansions will allow you access to higher levels and more areas. That Battle Chest for WoW is only for that, not for the ability to level the characters past 20. You still need to sign into your WoW account and purchase the monthly sub.