You need to use the correct install disk to fix your file system from it. However, in the absence of the correct install disk, there are two things you can try:
(1) Do a Safe Boot. Hold the Shift key when starting up. If done properly, the boot up will take a longer time and the screen will look different than usual, and it may show a different progress bar. It might take 5 or more minutes. Don't interrupt it. After it is done booting, then restart normally and try your El Capitan update again.
(2) Boot into single user mode and use the fsck -fy command. Details are at
Resolve startup issues and perform disk maintenance with Disk Utility and fsck - Apple Support
but here is the basic process from the Apple Support page referenced above:
(1) Start up your computer in single-user mode to reach the command line. This means, hold Command-S when starting up your computer. When doing this, a different user interface will appear, looking much like a typewriter bare bones interface, not the usual graphics user interface.
(2) Wait a minute until the lines of text stop scrolling. Wait an additional 30 seconds. At the command-line prompt type:
/sbin/fsck -fy
Press Return. fsck will go through five "phases" and then return information about your disk's use. Once it finishes, it'll display this message if no issue is found:
** The volume (name_of_volume) appears to be OK
If fsck finds issues and alters, repairs, or fixes anything, it displays this message:
***** FILE SYSTEM WAS MODIFIED *****
If this message appears, repeat the fsck command you typed in step 2 until fsck tells you that your volume appears to be OK (first-pass repairs may uncover additional issues, so this is a normal thing to do).
When fsck reports that your volume is OK, type
reboot
at the prompt and then press Return.
Your computer should now start up normally and allow you to log in.
At this point, retry your El Capitan update.