THE MIND OF A PSYCHOPATH #1
As much as his hybrids were his army, there is almost a pang of hurt over what he has to do. They committed a mutiny and everyone knows how a mutiny ends. Blood. Death. Murder. And a couple of heart extractions and decapitation. Klaus saw no other option. He held a kind of trust in his army to serve under him and for them to plot against him, with Tyler as their leader, he saw it as ungrateful. He cured them from having to go through excruciating pain every full moon and they prefer to plan on getting rid of their maker.
And Klaus watched as one of his hybrids run. They ran from their fate. But Klaus doesn't just speed in front of her. He simply walks, wielding his weapon and follows her. His face does not change composure because there is nothing for him to feel. This is the look of betrayal. This is no normal hunt. This gains no satisfaction to Klaus. This is simply karma. The only emotion to come out of this is when he angrily asks the girl where Tyler would be. And even her innocent reply was not satisfactory. He would have never spared her life. He wouldn't have just let her go after this. She was always going to die. As she gasps for her final breaths, Klaus just watches. This is when he holds her life in his hands and he knows what he is going to choose. There is no phasing on his face. Nothing more than emotionless as the head rolls to the floor.
Klaus had just killed what was left of what made him feel less alone. He was now alone again. And from this point on, his actions are run by this one singular place within him; loneliness.
When he finds Mayor Lockwood, he has clearly had something to drink, holding the bottle in his hand. His voice is masked with hurt, deceit and betrayal as he speaks to Caroline. He is hurt. Also Carol tries to persuade the hybrid, he just watches her. And that is when he kills her. He kills her because without her, Tyler doesn't have any kind of family. This is his final torture to Tyler. He kills his final family and maybe, just maybe, Tyler would know the kind of pain, the kind of loneliness that Klaus now feels.
Klaus didn't intend on going after Tyler. He never wished for that kind of energy. This was his final revenge. For Tyler to be alone and living in fear, even when Klaus was done with him. Much like Katherine. He slaughtered her family. He got his revenge upon her and yet, she ran in fear of him for centuries - even though he was not searching for her. If he was, she would have been dead. That is why Tyler is not dead.