For what reason has Mr. Martin decided to kill Mrs. Barrows?

Mr. Martin genuinely believes that Mrs. Barrows is planning to reorganize the company's filing department where he works. It's a perfectly reasonable assumption. Since wheedling her way into Mr. Fitweiler's life she's cut a swathe through F & S, turning the whole company upside down and generally getting on everyone's nerves with her ridiculous, meaningless jargon. Mr. Martin is at a loose end. Thoughts turn to murder. Thankfully, though, he realizes that there's more than one way to skin a cat. So at Mrs. Barrows's apartment one night, Mr. Martin makes up an incredibly tall story, saying that he's going to get high on heroin and kill Mr. Fitweiler.
It's a completely ludicrous story, but Mrs. Barrows is much too stupid and gullible to realize this. She also sees what she thinks is an unguarded comment as a great opportunity to remove Mr. Martin from the company and consolidate her power. So she spills the beans to Mr. Fitweiler. He may be old, but he still has all his marbles. The very idea of Mr. Martin doing such a thing is too ridiculous for words; Mrs. Barrows must be mad, he thinks. So he fires her. Mr. Martin is now in the catbird seat, and he didn't need to spill so much as a drop of blood to get there.