Determine how many times webpage has been accessed
Your server is already doing that, it is called the log file. If you grep it for the given page and count the number of log enteries, you will have the number of vistors who viewed that page. This, however, is inefficient and doesn't "work with CPanel".
Next, you could write a simple script to insert an entry to a database every time someone visits that page. Then your answer could be queried by custom PHP you add to CPanel. This is more effiecient, but would be reinventing the wheel.
But, the real answer is get analytics software. There are a number of products that can do this for you:
Piwik - This you can either get cloud-hosted, or host it yourself. It's free open-source software that does most of what Google Analytics does. You'd own your data and that data would reside on your server, there are widgets that could be embedded on pages so with some work could be integrated into CPanel. The reports are not as polished looking as Google Analytics, but that might be worth the trade-off.