There you go. Kirk probably thinks it might be Valeris during that conversation in the hall, but when they find the bodies of Burke and Samno, all doubt is removed from his mind.Kirk's suspicion is also set up just before they find the bodies:
SPOCK: I have reason to believe that Gorkon's murderers are aboard this vessel.
KIRK: I have a thought about that.
One thing that Shatner and Nicholas Meyer both understood: In a TOS story, Kirk has got to be the mover and shaker. Spock may come to his own logical conclusions and steer Kirk in a certain direction, but ultimately, the decision is the Captain's.