You can't justify killing one good person by killing one bad person.
What kind of screwed logic is that?
In DB you kill and there are no limitations.
Only because you kill some bad people does not justify killings of those good ones.

It's not necessarily "cold blooded" per se. Cold blooded means, generally, that you premeditated it, planned it out, and felt nothing. You could RP it as a crime of rage (e.g. you see how horrible she is and kill her in a fit of rage). Murder? Yep. Cold blooded murder? You could argue it was hot blooded murder. I'm not saying it's right or good, mind you. Murder is murder, but the game doesn't hand you a lot of options. My character seems to murder an awful lot of people in cold, warm, tepid, and scalding blood. Adding an abusive crone to the list so I could destroy the DB didn't seem like a huge stretch.

'You can burn her with a spell using Destruction.
She dies, but didn't really kill her directly, did you?
She just dies from the burns made by fire.'

You getting what I'm trying to point out here?

You don't have to be a cold blooded murderer. You could just be checking out the circumstances that caused the little tyke to run away... Then you talk to the children at the orphanage and find out about "the room" where they are shackled in for days without food. That caused my very good-hearted character to lose it, kill Grelod, and then dismantle the DB.