My computer will not start after update attempt

What is the error alert? Do you get to any screen at all? If so what screen?

What model of Mac do you have? From what version were you trying to upgrade from?

Can you boot into Safe Mode (How to use safe mode on your Mac - Apple Support). 

NOTE: Safe Mode boot can take up to 10 minutes as it's doing some system cache cleaning, volume verifying and directory repairing. 

If so then try rebooting normally.

If not try booting into the Recovery volume (boot with the Command + R keys held down). Select Disk Utility and run First Aid on the drive. Reboot normally.

If you still can't reboot normally boot again into the Recovery volume and select Reinstall OS X.

How is the Mac connected to the internet?

See if you can disconnect from the internet, i.e. unplug it from the router/modem or shut down the modem and then see if you can boot into the Recovery volume. Long shot but worth a try.

Have you tried booting into the Safe Mode or Recovery volume?