Grelod the Kind is one of those Skyrim characters that the entire community loves to hate. Being an abusive orphanage owner will do that to a person. Unlike other hated characters like Nazeem or Belethor though, you actually get to murder Grelod as part of a quest, as Aventus Aretino tasks you with killing her believing you're a member of the Dark Brotherhood.
Unfortunately, the quest itself is rather straightforward considering Grelod is your first kill before you begin the Dark Brotherhood questline, with the only real option to take her down being to sneak up and stab her. Thankfully, one modder has taken it upon themselves to give players a wider range of options to deal with Grelod, even one for all of you out there that wish to keep your hands clean.
The Innocence Lost - Quest Expansion mod created by JaySerpa can be found on Nexus Mods and introduces multiple new ways to take Grelod's life, including the ability to poison her favorite wine and push her into the ocean after luring her down to the Riften docks. You'll have to talk to Grelod to get these options though, and once you do bump her off, the Dark Brotherhood questline will resume as normal.
If you're roleplaying as more of a pacifist, instead of having to let the orphans continue to be abused by Grelod, you can actually have her arrested. Exploring the orphanage itself will allow you to collect evidence of Grelod's horrible deeds, or you can just use your power as the city's Thane to have the guards come and take her away. Once she's locked up, you can visit her to find that the Dark Brotherhood has fulfilled Arentino's request and killed her in jail.
Be aware though that going down the pacifist route actually locks you out of the Dark Brotherhood questline entirely. Not killing Grelod means that you don't give Astrid a reason to kidnap you, which means you never actually make contact with the Dark Brotherhood. Then again, if you aren't murdering someone as vile as Grelod, then you're probably comitted to the whole "no murder" thing.