To start playthrough 2 do the following:
Once you've killed the Destroyer, take the Fast Travel to the station "Underpass." Go to Tannis' and sell her the key to the vault. Once this is done, save and quit. Now, attempt to create a new game with that same character. When you're in the game lobby it will give you an option of playing Playthrough 1 or 2. (Doctorgray)
You can start Playthrough 2 before selling the key. Once the Destroyer is beaten, exit to the main menu. The game will autosave and there will be an option to start playthrough 2).