blue screen after interrupted software update

With the power adapter in use, not relying on battery alone when

attempting to upgrade or update software, is the first consideration.

Did you try startup in Safe mode to see if that gets you into it further?

• Try safe mode if your Mac doesn't finish starting up - Apple Support

You may have to try & start in OS X Recovery to see if the Disk Utility

and other tools available in there can help you get to a working OS X.

• About OS X Recovery - Apple Support

Either or both of the above should be able to get a rise out of the Mac.

Details are important, so if some aspect does not appear to work the

first time(s) to try them again carefully, is routine.

Expect to be asked to login to your admin account (password) at some

point after waiting to get into Safe mode. And later on, if you succeed,

in order to access the full system, you'd need to restart Mac, normally.

Hard to say what to try at some distance. If you are nearer an Apple store

(official stores have Genius help, by appointment) you could see about the

help they may be able to provide. Authorized Apple Support Providers and

Resellers may be able to help, their trained service techs get customary fee

based compensation for their shop time or sometimes may help w/o charge.

• Make an appointment at an Apple Retail Store Genius Bar
• Find an Apple Authorized Service Provider

If your Apple product has AppleCare plan in effect, that may be helpful if

your choices are limited; with official Apple store, a genius appointment may

be a fair way to see if they can help; or they may refer you.

In any event...

Good luck & happy computing! 🙂