Grandpa is our third narrator, and we get to know him through Oskar's descriptions and a series of letters he's written in his notebook but never sent. Born Thomas Schell, he lived through the bombing of Dresden, in which he lost his entire family. As a young man, he was madly in love with Anna, who became pregnant and was killed, along with their unborn baby, in the firestorm. After the bombing, he had to kill a bunch of animals that escaped from the zoo. He's a sensitive and artistic man. He likes to sculpt and, probably out of guilt, he keeps so many animals in his apartment we're surprised it hasn't been featured on Pet Hoarders.
As a result of his trauma, he gradually stops talking and communicates mainly through gestures, writing notes, and the words YES and NO tattooed on his palms. He meets and marries Anna's sister in New York, even though he doesn't speak. For the both of them, that's a bonus:
She never had to know me. […] We'd both come to New York lonely, broken and confused. (2.1)
They seem to marry just for the companionship. Grandpa can't stop thinking about the past, but Grandma acts like the past never happened. It's a good match.
When Grandma breaks one of their marital rules by getting pregnant, Grandpa leaves. "Why couldn't I be the kind of person who stays?" (6.1). He seems to be resigned to this, as though it's his fate to run away. He can't overcome the trauma of the loss of Anna.
He spends years writing to his unborn child, a child who can be either Thomas Jr. (Oskar's Dad) or the baby that Anna was pregnant with when she died in the bombing. But Grandpa only sends one letter, a letter that Thomas marks up in red pen. Grandpa only meets his son once, when he came to Europe posing as a journalist. Neither admitted to the other who he was. He returns to New York shortly before Thomas Jr. dies in 9/11.
That's when Grandpa becomes "the renter." At least, that's how he's known to Oskar. Grandma doesn't want Grandpa to get to know his grandson (probably because of that whole abandonment thing) so he's only known as "the renter" to young Oskar.
They still bond in their own way, though. The renter helps Oskar dig up Thomas Jr.'s grave. This gives both Grandfather and Grandson closure, although more for Grandpa than for Oskar. Grandpa's able to bury all the letters he wrote and feels, in his own way, like they were finally delivered.
Oskar feels a kinship with the renter even before he realizes he's his Grandpa. Grandpa's a great listener (by default, we guess) and is described as serious and mysterious. The picture we get of him through Oskar's descriptions and the one that emerges in his letters couldn't be more different. He has a lot to say in those letters.
Grandpa falls back on his old habit after the un-burial: running away. Except this time, Grandma follows him to the airport and convinces him to stay… at the airport. Yes, it seems the two live at the airport at the end of the novel. They both realize they can't decide whether to move forward or backward, so they stay somewhere in between. Grandpa says he likes to "see people reunited" at the airport, but can't seem to bring himself to do that.