Like others have said, definitely do deadlifts, they hit critical muscle groups for posture. Which muscles? Those along the spine from the mid to upper back; they're "dead" for a lot of people thanks to years of slouching, but those are the muscles that can lift your chest+head up while letting the shoulders relax into a natural and wider position, instead of looking stiff. Elliot Hulse provides a great technical overview of how to use deadlifts to build thoracic extension for better posture (esp. starting around 2 minutes in):
http://youtu.be/9BTQebgxi8A
Another exercise that hits that area but does not require a gym is the "cobra" or "floor cobra" - there are different versions from yoga, Pilates, etc. and here's one:
http://youtu.be/lODeZLnxZgQ
To self-diagnose and check progress, regularly stand against a wall: put your back against a wall, pull heels together all the way in to touch the wall, and try to flatten your upper back (not including shoulders) and back-of-head against the wall, keeping chin down and eyes level. Then step away from the wall; it will feel strange at first, but if you're near a mirror, you should catch yourself in near-perfect posture: from the side, your ears, shoulders, hips, and ankles will be in a straight line down (or pretty close).
http://youtu.be/9BTQebgxi8A
Another exercise that hits that area but does not require a gym is the "cobra" or "floor cobra" - there are different versions from yoga, Pilates, etc. and here's one:
http://youtu.be/lODeZLnxZgQ
To self-diagnose and check progress, regularly stand against a wall: put your back against a wall, pull heels together all the way in to touch the wall, and try to flatten your upper back (not including shoulders) and back-of-head against the wall, keeping chin down and eyes level. Then step away from the wall; it will feel strange at first, but if you're near a mirror, you should catch yourself in near-perfect posture: from the side, your ears, shoulders, hips, and ankles will be in a straight line down (or pretty close).