If you're still reading, that means you're ok with the fact that we're about to talk about pivotal plot elements which happen in Logan. As we noted in our review of the film, unlike the earlier entries in the X-Men franchise, here the story pulls no punches when it comes to dealing with life and death.
In the original Old Man Logan comic storyline, Wolverine had fought a rampage of super-villains who were attacking Xavier's School, killing them. However it turned out that the villain Mysterio had put that image into Wolverine's mind. It was actually the other students and inhabitants of the school that Wolverine had killed. While that is not directly stated during the film, Logan, the discussions between Logan and Charles are such that it seems likely (or something like it). It's why there are no other good mutants left around, and why Logan is drinking himself into a daily coma.
The adamantium infused in his bones over the decades has been slowly leeching into his body, causing heavy metal poisoning, which has always been the issue – just his healing abilities have been strong enough to keep it at bay. Now his body is exhausted and it can't keep up. Now the metal in his body is killing him, and there's nothing that would be likely able to reverse it. His body, like the world around him is worn down and giving up.
The illegal experiments that have been going on down in Mexico have involved developing a new generation of mutants, not by finding individuals that are manifesting their powers, but by creating clones of known mutants and pushing them until their abilities reveal themselves. Laura is a partial clone of Logan, which is where she gets her reflexes, claws, and her healing powers; her own adamantium infused bones were courtesy of the scientists that were keeping her captive. When the culmination of the program comes online, X-24, their captives decide to kill the children off, with only some of the nurses fighting to try to help them escape.
While Laura turns out to be the biological equivalent of Logan, X-24, when he appears in the second half of the film, is an outright clone. He's Wolverine in his classic prime with all of the anger and violence, but none of the humanity. Logan in the condition he is in the film would never have been able to beat a younger version of himself alone. I
Yes, the film includes the deaths of both Professor X/Charles Xavier as well as Wolverine/Logan. They're not pretend dead, or kinda dead, they're just dead. There's always the possibility that Logan will somehow spark his regeneration (either internally or from some external influence), however if they just turn around and bring him back that would cheapen this film immeasurably.
With Xavier, he is in his bed, finally having had a night of family, home, and lucidity one last time. Only to feel a presence in the room, seeing Wolverine, who uses his claws to mortally wound Charles. It turns out to be X-23, and there's another of the film's heartbreaking moments when we realize that Charles may have thought that he had just been killed by his old friend. There is a moment just later when Logan has found Charles and is repeating to him over and over "it wasn't me." It's not clear if Charles heard him, or understood if he did. One more burden to add to Logan's mountain guilt.
The final scene with the children, leaving for Canada (presumably to meet up with a kinder-gentler government, possibly to form a new version of Alpha Flight), opens with them standing around Logan's grave. Laura recites the scene from the film she watched with Xavier earlier in the casino hotel room. It's from the '53 western, Shane, and it speaks of the main character's inability to escape what he's done, so he has to find a way to live with it, and accept it.
There's no living with… with a killing. There's no going back from one.
There's a moment of silence with the children standing there but it is as much for the audience, for whom it seems that we are saying goodbye to a character that himself finally came to terms with his past, and became a leader for these children, giving his life so that they could escape into a better life than he had. As they file out, only Laura remains, finally lifting the simple cross they had fashioned out of sticks out of the ground and laying it on it's edge, forming an "X". With the passing of the last of the X-Men, she turns into a clouded, but hopeful future.