Try booting the computer while holding Command + R. This should boot the computer into the recovery partition. From there you can open Disk Utility and check the health of your Hard Drive.
Once you open Disk utility select your hard drive and click the First Aid tab. Then click verify disk, if this comes back red your upgrade to El Capitan may have gotten corrupt. If you have a backup awesome, if not you can try to backup through target disk mode. You are able to reinstall Mac OSX through the recovery partition as well, you do not want to reformat your disk just install. The install should not touch your data but I recommend you have a backup first either way.