This is book 2 of the "Hornblower" series.

This note contains spoilers for the book.

He meets Bush on the first page. Bush turns out to be senior to him, but they're both junior lieutenants on the Renown.

The Captain is crazy, in a paranoid way, and has an unfortunate fall into a hatchway that eventually kills him. Nobody ever does find out how it happened (though there's a slight suspicion that Hornblower may know).

Then, with Buckland in Command and Hornblower sometimes pulling the strings a bit, they have to try to carry out their mission.