Here the planet looks about the size the earth does from the moon, and since Alderaan is about the same size as the earth, I would guess, about the distance the earth is from the moon, maybe a little farther. It does appear to have moved farther away from the destruction by the time the Millenium Falcon arrives, as the Falcon probably came out of hyperspace only a few hundred thousand kilometers away from where the planet was, maybe even closer and the distance from the debris to the Death Star seems considerably longer. I was wondering, though, is the distance canonically established, and did the Death Star purposefully move away to avoid getting hit by debris from the planet?
They were very close indeed. Assuming Alderaan is roughly the size of the Earth, they were only about 50,000 miles away, less than 20% the distance of the Earth to the Moon.
A voice announced over a hidden speaker that they had approached within antigrav range of Alderaan — approximately six planetary diameters. That was enough to accomplish what all of Vader’s infernal devices had failed to.
And yes, I'm fully aware that that's dumb. Don't yell at me.
All I know is that the death star was not significantly damaged by the explosion of Alderaan. Thus it was always far enough away, perhaps moving away slowly all the time, not to be damaged.
Perhaps someone will be able to calculate a minimum safe distance for the Death Star from the degree of vulnerability of star destroyers to asteroids etc., and possibly someone might be able to calculate a maximum possible distance from the scene in the film - a novelization suggests a distance of about 50,000 nukes as stated by Valorum above. And we can HOPE that the calculated minimum safe distance will be less than any calculated maximum possible distance.