This is a short list of books I would personally recommend. Most have lengthy bibliographies to take you further. As an adult educator I try to choose books I think you would enjoy and benefit from reading rather than books universities place on student booklists because they believe they should read them!
General Books
Continue to Pester, Nag and Bite Martin Gilbert
Finest Years Max Hastings
Last Hope Island Lynne Olson
The Second World War Winston Churchill
Taking Office May 1940
Six Minutes in May Nicholas Shakespeare
Cabinet's Finest Hour David Owen
1940 Laurence Thompson
Blitz and Home Front
The Oaken Heart Margery Allingham
Blitz MJ Gaskin
The Blitz Juliet Gardiner
The Splendid and The Vile Erik Larson
and on 'the Churchill Industry': The Churchill Myths by Steven Fielding et al
There are thousands more books you could read and prosper from!