I redeemed my DLC code, but nothing is showing up in-game. Where is it?

Once you have entered the actual redemption code, such as the retailer exclusive preorder DLC pack, from in-game simply press Menu (Start), select Options and then General Options.  In the lower-right corner of the screen, you will see the option to redeem DLC/bonus content.

From there, you can activate each item individually.  Please note that avatar customization options can only be equipped and switched out at a space station's Appearance Modifier, so you will have to progress to a certain point in the game before those become available.