How to start a second playthrough with an existing character?

I completed my first playthrough without reaching level 50 or 60 (there are achievements for attaining both of those levels).

Yes, there are unlocked achievements for both of the levels, namely, Power Gamer and Extreme Power Gamer. See here

To continue a completed game, just choose new game and you will see your previous character there. Simply select it.

Thanks yx_ for your answer. The exact menu choices are as follows:

1. From the Main Menu choose "Start New Career".
2. Hit the "A" button to "Connect to Database". (The game will establish a secure connection and take you to the Profile Reconstruction Menu.)
3. Choose "Select Existing ID" on the Profile Reconstruction menu.
4. Select your existing character from the list of available careers. (Careers are listed with the characters name and level.)