Here are my 2 cents:
Control charts and Cp and Cpk figures are closely related, but some differences are important to notice.
First: Data set for your capability analysis must follow a normal distribution. It is an underlying assumption.
Second: Control charts assumes normal distribution when de process is stable. If you are using an Xbar / R chart, with a sample size of 4 or 5 at least and your process is stable, then you can be confident with regard to the normality distribution of your data (There is a statistical theorem behind this). Individual and Moving Range charts are more sensitive to departures from normality, therefore is advisable to check for normality (even if your process is stable)
Practical conclusion: If your chart is an I/MR one, then the normality in the control chart is enough for the Cp and Cpk indexes to be valid, but, if your chart is an Xbar/R one, then you must check for normality of your data before analysing the capability indexes.
Hope this helps.