We first encounter Grandfather when Arthur talks to him remotely about the food-gathering trip aboveground. Grandfather cautions Arthur to be careful, and also reminds him:
"And remember, only take from the bigger gardens—and only then if they have plenty! There are a lot of people who can only survive by growing their own food." (1.8)
Sounds like Grandfather is pretty compassionate, which is lucky for Arthur since Grandfather found him "abandoned on the steps of the poorhouse" (5.56) when he was a baby. Grandfather took him in and raised him like a son, which if you ask us, is a pretty obvious indication that he's a nice person.
Grandfather does have his temper, though. When Arthur calls him on the doll-communicator to report back, and says that he's in the Cheese Hall, Grandfather responds by saying:
"WHAT? You're in the Cheese Hall? […] Why can't you do what you're told? I'm very cross with you." (25.8-10)
We can't blame him for getting upset, though, since Arthur is like family to him, and it's clear that Arthur could be in danger. You don't find a baby and raise them as your own just to check out when they grow up a bit and start making reckless decisions, after all. So Grandfather can have a bit of a temper, but we consistently see it used for good, which is fitting for "a stocky old man with a huge beard and glasses" (40.4). After all, it sounds like he looks a little like Santa, and nobody likes a mean Santa.
Whenever Grandfather talks to Arthur via the doll-phone, he tells Arthur to be cautious. For instance, when Arthur is about to check out the cheese hunt he sees, Grandfather says, "'Well, keep up on the roofs, and see if you can follow them… But… be very careful!" (3.18). Okay, we get it: Caution is a priority.
The whole caution-thing only gets more intense when Grandfather finds out that Snatcher is involved in this whole mess. He tells Arthur via doll-phone:
"Listen to me, Arthur. You are not to do anything rash. That is an order! I don't want you to do anything but watch. Mr. Archibald Snatcher is a very dangerous man!" (19.15)
We picture Grandfather saying this in his most serious of serious voices. But in case Arthur isn't picking up on just how much Grandfather means it, before Arthur gets off the doll-phone to go investigate something happening outside the Cheese Hall, Grandfather reiterates his warning to be cautious: "'Arthur! Arthur! Be careful!'" (19.22). It sounds like Grandfather is really spooked by the possibility of Arthur getting tangled up in all this stuff, though we'll have to wait until later to find out why.
Okay, so we've gathered that Grandfather is pretty crafty. He somehow invented wings and a doll that's actually like a walkie-talkie for Arthur. So he has a background in engineering or something, right? Question mark totally intended, since despite living with the guy for pretty much his whole life, Grandfather's past is all a mystery to Arthur.
In fact, it takes a conversation with Willbury for Arthur to get his first hint about why he and Grandfather live underground. Grandfather says to Willbury:
"I was accused of a crime that I did not commit. I have had to take refuge here ever since." (10.26)
Whoa, that's intense. But instead of giving us much information about why Grandfather is underground, our curiosity is piqued and we find ourselves with a whole lot more questions. What the heck was Grandfather's life like before he moved underground?
Later, Grandfather reveals to Arthur that Snatcher "'is the reason we live down here!'" (19.17), but it isn't until Arthur lands in the dungeon under the Cheese Hall and finds himself locked up next to Grandfather's childhood buddy Herbert that we get the whole story, which goes a little something like this: Snatcher framed Grandfather for attempted murder, and the only thing Grandfather could do was run. Since Herbert had disappeared (locked up by Snatcher), Grandfather just stayed underground. Grandfather says:
"I grabbed some food from a garden, then went back underground to avoid being caught." (39.55)
By the end of the story, Grandfather is on his way to finally having his name cleared. He is also returning to life above ground, which is a long overdue move. Because of this, he and Arthur get to live happily ever after. Yay.