Take a look at this article on the "Third World Squat". It will help you identify what the problem areas are from sitting all day.
The problem we see is this:
- Lack of strength in the posterior chain. You aren't using these muscles while sitting in a chair.
- Lack of flexibility in the hip flexors, calves, hamstrings
- Too much flexibility in the glutius maximus.
If you want to stick with body weight exercises, which would be enough to correct this:
- Start with back extensions (also known as "hypers" or hyperextensions).
- Next you'll need to start stretching out the problem areas.
From this point you have a couple paths you could go down to maintain good posture and the additional strength you've accumulated to support it. You could either go down the strength training route, or you can go down the pilates route. Pilates provides a combination of body weight exercises designed to improve both flexibility and strength. Strength training provides the ability to strengthen your muscles to the point where slouching is just not possible. It really depends on what suits you best.