I'm not by any means read up on the books. I've been following behind to experience the series first and the books afterward.
Granted, the series is making its own changes as it goes, but from everything I've seen from the book readers, it's still quite faithful to the overall flow. And many key scenes seem to be right off the pages from the descriptions that I've read.
For example, the Battle of the Blackwater was thrilling in the episode. As Martin noted in his commentary on the episode, they simply could not afford to also include the chain. I understand that it's Tyrion's masterstroke in trapping Stannis' fleet in the situation, but I honestly didn't miss it in the episode.
The depiction of Tyrion's final acts in the last episode did not play for me as the behavior of a cold-blooded or vicious man. Had they played a situation of Tyrion verbally attacking Jaime, swearing revenge on him and falsely confessing to the poisoning, that would have been a completely different scenario. The series never made a real character out of Tyrion's first wife. She was mentioned at several points along the way, with it clearly being a sore spot between Tyrion and Tywin. But the series clearly has wanted to play the relationship between Tyrion and Jaime as a positive one - they wanted them to part as friends, not in anger.
I did not get the impression in the series that Tyrion went into the Tower of the Hand expressly to kill his father. I believe he went there to confront him before leaving town. I do think he wanted an answer from his father how he could have sentenced him to death for a crime he didn't commit. The discovery of Shae in Tywin's bed was a giant slap in the face. It shows that his father was not only a hypocrite but had been sleeping with Shae the whole time Tyrion was locked in the dungeon. It potentially shows that Shae's lies in court were encouraged by his father to get him sentenced to death. It means that his father really did want him dead and was taking advantage off the convenient Joffrey situation to get it to happen. (It also seems to me that Tywin would have been fine with the "deal" he was making with Jaime - to have Tyrion banished to the Night's Watch rather than outright killed. Once Tyrion challenged that in court, Tywin was happy to have Tyrion indeed executed)
The relationship of Shae to Tyrion in the series plays differently than the cunning person she is described as in the books. The series played it that both Tyrion and Shae loved each other. (I'd argue that they were making a case for on-camera scenes playing better than a description of a woman Tyrion loved many years prior, who we never actually see.) In the show, Tyrion takes multiple steps to protect Shae from Cercei and Tywin, who he believes could have her killed. When he is at his most desperate to get Shae out of the area, he says hurtful things to her that he doesn't mean, hoping this will convince her to leave King's Landing and have a safer life somewhere else. Shae is presented in the show as not being that intelligent, since she really doesn't understand the danger she's in, or what he's trying to do for her. Given this, Shae feels completely rejected by Tyrion. He's already rejected her advances multiple times since marrying Sansa, and now he's, to her thinking, tossed her out on her ear. Her betrayal in court comes across as the act of a spurned lover rather than a calculated move. And the discovery of her in Tywin's bed indicates that Tywin has taken advantage of her "rejection" by Tyrion and used it to help himself. When Shae sees Tyrion in Tywin's bedroom, an argument can be made that she thinks he's there to kill her. (The description of what happens in the book seems to bear that idea out.) She goes for the knife, and the whole situation goes badly from there. I don't know watching the scene that Tyrion would have killed her in the show had she not grabbed the knife and attacked him. It absolutely plays as self-defense, at least at first. It pushes past self-defense when Tyrion kills her rather than just incapacitating her. That's where the anger and the hurt really play.
I agree that in the show, once Tyrion has killed her, he's fairly set on killing his father. Once Tywin sees Tyrion with the crossbow, I think he knows he's toast, but he thinks he can talk his way out of it. He offers Tyrion the compliments and reassurances he thinks Tyrion wants - that he respects and admires Tyrion, that he'd never let him be executed, that he acknowledges Tyrion as his son. And of course none of this is working now because Tyrion has Shae's blood on his hands and is fully aware that Tywin was orchestrating not just banishment but his death. Tywin's reference to Shae as a whore, after Tyrion has stated his feelings about her, is the last straw. And Tywin can't answer Tyrion's charge that he was ready to see Tyrion condemned to death for a crime he didn't commit. After Tywin rubs more salt into the wound by calling Shae the name again, Tyrion's done. And so is Tywin, who finally shows his true colors when on the verge of death, saying "You're no son of mine".
I realized after the show that this scene was the only time we'd ever seen Tywin out of his uniform. Every other time, we would see Tywin in his full red and gold regalia, looking quite dignified. For this scene in the privy, Tywin was seen out of his armor. In a sense, we and Tyrion are seeing Tywin for who he really is. And I did not believe his statements about respecting Tyrion or that he wouldn't have him killed. I think the second Tywin got out of that privy and got back to his chambers, he'd have a guard take Tyrion away immediately. Or just take his head. Only one of those two men was going to walk out of that situation alive.
Granted, none of this reads quite the way the situation is being described for the book. But I think for the show it's compelling to have the key relationship for Tyrion be one that we can see happening in front of us. In a book, we can visualize many other parts of the characters' lives without having an immediacy to them. In a movie, play, or TV show, we need to see those relationships play out. Yes, a monologue by Tyrion about the cousin killing all the beetles is a revealing thing - it tells us about Tyrion's reaction to what he was seeing. But that wouldn't be enough to show us on camera why Tyrion takes the actions that he does here. It would be akin to removing the Sicily scenes of Michael Corleone from The Godfather but basing his ruthless behavior on the death of his wife there. It's a lot more compelling to show the Sicilian wife to understand why she's important. Just talking about her would be fine in the book. In a movie, it's just that - talk.
This is all a fairly long-winded way for me to say that I think they found an understandable way to present Tyrion's actions in terms of what we've seen on screen. And yes, it keeps a little of the "good" in him throughout - we can empathize with his anger and hurt. But he still does kill these people. I suspect in the coming years, we won't be seeing a lot of the happier, more jovial Tyrion we saw in earlier seasons with Bronn and Pod...