Hi Courtney_22. Thanks for posting your question.
Your desire to go from a 10B to a 10C and to achieve a natural looking result is very sensible. If you have non-ptotic, reasonably good breast shape to begin with, then I think round implants placed under the pectoral muscle will give you a good result.
In terms of sizing, it is difficult to tell you how many CCs will give you a definite resultant cup size, or even the appearance that you desire. Keep in mind that two people of the same cup size may have quite different looking breasts, since many other factors, such as chest size and shape, can play a role in bra size. I have found the best way to determine implant size in breast augmentation is to have the patients "try" on the various implant sizes by placing implants or sizers under the bra. Having the ability to both SEE and FEEL the change can better lead you toward your size of choice. Based your breast dimensions, your plastic surgeon will be able to determine the range of implant sizes that your body can safely accomodate.
All the very best.
Dr Ray Goh 

Courtney,
the implant size that will achieve a certain result will depend on your frame and the dimensions of the implant.
I would suggest using a sizing system with your surgeon to have an idea of the volume needed to achieve your desired result, and then use the dimensions of your chest to determine the style of implant needed.
I hope that is of some help
Jeremy Hunt