Genovese had desired the top spot for a very long time. And when you look at Genoovese´s original plan of getting rid of Costello, it seems simple but Machiavelian at the same time. Costello was not a threat to Genovese. Anastasia (who was a close friend of Costello and his muscle) definitely was.
But taking over the top spot from Costello by first killing Anasatsia would not have accomplished anything. The Commission would had been all over him and perhaps even punished him for killing a member of another Family without permission. But by killing Costello first and claiming self defense is somehow more justifiable in front of the Commission. After all, it had been done before. Members of the Commission, Lucchese and others who were allied with Genovese, would feel that defending Genovese would be the right thing to do. "Vito was only protecting himself..."
The Commission´s ruling would be essential. According to plan, it would go in Genovese´s favor and the ruling would have pacified Anastasia, the dead Costello´s friend. It would also have prevented him from demanding revenge.
Now, as it turned out, Costello survived the attempt on his life. He was still alive and very much able to confront Genovese´s accusations.
Here is when Genovese "fled" to his mansion with 40 or so men, and Genovese soldiers started to hit the mattresses. Genovese feared Anastasia´s wrath. Now was the time to fear war, now when Costello was still alive.
War never erupted though. Because majority of the Commission members sided with Genovese anyway and decided for the sake of peace (and other foolish transgression by Anastasia that had amounted) to get rid of Anastasia instead.

So it was safer for Genovese to kill Costello off properly than just sending him a warning. Are you following my logic, Danny?