the same character has to kill Grelod the not so kind?

Depends on how you rationalize your kills. Grelod is a cruel woman, killing her might be seen as justice but what happens to the kids afterwards? You'd have to weigh that as well. And then the next portion of the quest, would a morally good person execute someone on the say so of someone they don't know? If not would they instead attack the unknown assailant that brought you to the abandoned shack? It's all in how you rationalize it. Morals are flexible given the proper motivation, how morally astute is your character, can they be persuaded to do something they think is wrong, if your character is adverse to violence they likely wouldn't kill even for a good reason, if they are alright with violence to what extent can they push their own moral boundaries. Will your morally good character start down the dark path, justifying his actions with "the greater good" or will he refuse to dirty his hands. If he does head towards the dark will he eventually keep making excuses until he doesn't know right from wrong and instead becomes the thing he despises? All good questions. He's your character, you decide.