The methods used by chromatography data systems are usually proprietary, so very little information is available to the public. However, I have done similar programming for simple applications and I can give you some ideas.
To get the peak area you must know the starting point and ending point for the peak. These points can be found by examining the first and second derivatives of the signal, and finding the point where the derivative exceeds some "threshold" amount, which is arbitrarily set.
The retention time is usually defined as the maximum point on the signal, which is easy to search for. You could also estimate it by using derivatives, or doing some curve fitting (second order polynomial works well) for the points near the apex.
The area is usually calculated as a histogram summation from the integration baseline to each data point, remembering to account for the data acquisition rate, which is set in the acquisition parameters.
This should get you started.